First let's find the roots and factor $g(\omega)$ so the whole denominator of the integrand is factored completely. We'll need it factored to find residues anyway.
Completing the square to find the roots:
$$\begin{align*}g(\omega) = \alpha\omega^2 + i\beta\omega + \gamma &= 0\\
\\
\omega^2 + i\dfrac{\beta}{\alpha}\omega + \dfrac{\gamma}{\alpha}&= 0\\
\\
\omega^2 + i\dfrac{\beta}{\alpha}\omega - \left(\dfrac{\beta}{2\alpha}\right)^2 &= - \left(\dfrac{\beta}{2\alpha}\right)^2-\dfrac{\gamma}{\alpha}\\
\\
\left(\omega + i\dfrac{\beta}{2\alpha}\right)^2 &= - \left(\dfrac{\beta}{2\alpha}\right)^2-\dfrac{\gamma}{\alpha}\\
\\
\omega_{1,2} &= - i\dfrac{\beta}{2\alpha} \pm \sqrt{- \left(\dfrac{\beta}{2\alpha}\right)^2-\dfrac{\gamma}{\alpha}}\\
\\
\omega_{1,2} &= i\left[-\dfrac{\beta}{2\alpha} \pm \dfrac{\sqrt{ \beta^2+4\alpha\gamma}}{{2\alpha}}\right]\\
\\
\end{align*}$$
Thus
$$g(\omega)=\alpha\left(\omega-\omega_1\right)\left(\omega-\omega_2\right)$$
and I'll stop using the $g(\omega)$ notation.
You are interested in integrals of the form
$$P.V. \int_{-\infty}^{\infty}{\dfrac{ie^{i\omega t}}{\alpha\left(\omega-\omega_0\right)\left(\omega-\omega_1\right)\left(\omega-\omega_2\right)}}d\omega$$
With $\alpha$, $t$, $\omega$, and $\omega_0$ real; and $\omega_1$ and $\omega_2$ complex.
I am specifically excluding cases where $\gamma$ is $0$, as that puts another singularity on the real axis (either $\omega_1$ or $\omega_2$ becomes 0) that needs to be handled specially. You can extend what I do later in this answer to handle that case, if you desire.
Consider the complex contour integral
$$\oint_{C}{\dfrac{ie^{iz t}}{\alpha\left(z-\omega_0\right)\left(z-\omega_1\right)\left(z-\omega_2\right)}}dz$$
This contour integral, the Residue Theorem, Jordan's Lemma, and a properly selected contour can be used to find the Principal Value of your improper integral on the real line.
You need to consider cases for $t >0$, $t<0$, and $t=0$, as they require different contours or different handling altogether.
For $t>0$, select closed contour $C_P$, that is a semi-circular contour in the upper half plane, that includes the real axis as its straight edge with a small semi-circular arc excursion into the upper half plane around $\omega_0$.
From the Residue Theorem
$$\begin{align*}&2\pi i \sum_{\omega_k \in \mathrm{UHP}} {\underset{z=\omega_k}{\mathrm{Res}}\left[\dfrac{ie^{iz t}}{\alpha\left(z-\omega_0\right)\left(z-\omega_1\right)\left(z-\omega_2\right)}\right] } = \oint_{C_P}{\dfrac{ie^{iz t}}{\alpha\left(z-\omega_0\right)\left(z-\omega_1\right)\left(z-\omega_2\right)}}dz\\
\\
&=\lim_{{r \to 0},{R \to \infty}}{\left[\int_{-R}^{\omega_0-r}{\dfrac{ie^{i\omega t}}{\alpha\left(\omega-\omega_0\right)\left(\omega-\omega_1\right)\left(\omega-\omega_2\right)}}d\omega +\\
\\
\int_{\omega_0+r}^{R}{\dfrac{ie^{i\omega t}}{\alpha\left(\omega-\omega_0\right)\left(\omega-\omega_1\right)\left(\omega-\omega_2\right)}}d\omega +\\
\\
\int_{\pi}^{0}{\dfrac{ie^{it\left(\omega_0+re^{i\omega}\right)}ire^{i\omega}}{\alpha\left(\omega_0+re^{i\omega}-\omega_0\right)\left(\omega_0+re^{i\omega}-\omega_1\right)\left(\omega_0+re^{i\omega}-\omega_2\right)}}d\omega +\\
\\
\int_{0}^{\pi}{\dfrac{ie^{itRe^{i\omega}}iRe^{i\omega}}{\alpha\left(Re^{i\omega}-\omega_0\right)\left(Re^{i\omega}-\omega_1\right)\left(Re^{i\omega}-\omega_2\right)}}d\omega\right]} \\
\\
&= P.V. \int_{-\infty}^{\infty}{\dfrac{ie^{i\omega t}}{\alpha\left(\omega-\omega_0\right)\left(\omega-\omega_1\right)\left(\omega-\omega_2\right)}}d\omega \space+\\
\\
&\lim_{r \to 0}{\left[ \dfrac{e^{i\omega_0 t}}{\alpha}\int_{0}^{\pi}{\dfrac{e^{itre^{i\omega}}}{\left(\omega_0+re^{i\omega}-\omega_1\right)\left(\omega_0+re^{i\omega}-\omega_2\right)}}d\omega \right]}+\\
\\
&\lim_{R \to \infty}{\left[\int_{0}^{\pi}{\dfrac{ie^{itRe^{i\omega}}iRe^{i\omega}}{\alpha\left(Re^{i\omega}-\omega_0\right)\left(Re^{i\omega}-\omega_1\right)\left(Re^{i\omega}-\omega_2\right)}}d\omega \right]}\\
\\
&= P.V. \int_{-\infty}^{\infty}{\dfrac{ie^{i\omega t}}{\alpha\left(\omega-\omega_0\right)\left(\omega-\omega_1\right)\left(\omega-\omega_2\right)}}d\omega \space+\dfrac{\pi e^{i\omega_0 t}}{\alpha\left(\omega_0-\omega_1\right)\left(\omega_0-\omega_2\right)}\space + 0\\
\end{align*}$$
So for $t>0$
$$\begin{align*}&P.V. \int_{-\infty}^{\infty}{\dfrac{ie^{i\omega t}}{\alpha\left(\omega-\omega_0\right)\left(\omega-\omega_1\right)\left(\omega-\omega_2\right)}}d\omega \\
\\
&= -\dfrac{\pi}{\alpha}\dfrac{ e^{i\omega_0 t}}{\left(\omega_0-\omega_1\right)\left(\omega_0-\omega_2\right)} - \dfrac{2\pi}{\alpha}\sum_{\omega_k \in \mathrm{UHP}} {\underset{z=\omega_k}{\mathrm{Res}}\left[\dfrac{e^{iz t}}{\left(z-\omega_0\right)\left(z-\omega_1\right)\left(z-\omega_2\right)}\right] }\\
\end{align*}$$
And, for example, for $t>0$, if both $\omega_1$ and $\omega_2$ are in the upper half plane, then working out the residues, one gets
$$\begin{align*}&P.V. \int_{-\infty}^{\infty}{\dfrac{ie^{i\omega t}}{\alpha\left(\omega-\omega_0\right)\left(\omega-\omega_1\right)\left(\omega-\omega_2\right)}}d\omega \\
\\
&= -\dfrac{\pi}{\alpha}\dfrac{ e^{i\omega_0 t}}{\left(\omega_0-\omega_1\right)\left(\omega_0-\omega_2\right)} - \dfrac{2\pi}{\alpha}\dfrac{ e^{i\omega_1 t}}{\left(\omega_1-\omega_0\right)\left(\omega_1-\omega_2\right)} - \dfrac{2\pi}{\alpha}\dfrac{ e^{i\omega_2 t}}{\left(\omega_2-\omega_0\right)\left(\omega_2-\omega_1\right)}\\
\end{align*}$$
\
For $t<0$, select closed contour $C_N$, that is a semi-circular contour in the lower half plane, that includes the real axis as its straight edge with a small semi-circular arc excursion into the lower half plane around $\omega_0$.
From the Residue Theorem
$$\begin{align*}&-2\pi i \sum_{\omega_k \in \mathrm{LHP}} {\underset{z=\omega_k}{\mathrm{Res}}\left[\dfrac{ie^{iz t}}{\alpha\left(z-\omega_0\right)\left(z-\omega_1\right)\left(z-\omega_2\right)}\right] } = \oint_{C_N}{\dfrac{ie^{iz t}}{\alpha\left(z-\omega_0\right)\left(z-\omega_1\right)\left(z-\omega_2\right)}}dz\\
\\
&=\lim_{{r \to 0},{R \to \infty}}{\left[\int_{-R}^{\omega_0-r}{\dfrac{ie^{i\omega t}}{\alpha\left(\omega-\omega_0\right)\left(\omega-\omega_1\right)\left(\omega-\omega_2\right)}}d\omega +\\
\\
\int_{\omega_0+r}^{R}{\dfrac{ie^{i\omega t}}{\alpha\left(\omega-\omega_0\right)\left(\omega-\omega_1\right)\left(\omega-\omega_2\right)}}d\omega +\\
\\
\int_{-\pi}^{0}{\dfrac{ie^{it\left(\omega_0+re^{i\omega}\right)}ire^{i\omega}}{\alpha\left(\omega_0+re^{i\omega}-\omega_0\right)\left(\omega_0+re^{i\omega}-\omega_1\right)\left(\omega_0+re^{i\omega}-\omega_2\right)}}d\omega +\\
\\
\int_{0}^{-\pi}{\dfrac{ie^{itRe^{i\omega}}iRe^{i\omega}}{\alpha\left(Re^{i\omega}-\omega_0\right)\left(Re^{i\omega}-\omega_1\right)\left(Re^{i\omega}-\omega_2\right)}}d\omega\right]} \\
\\
&= P.V. \int_{-\infty}^{\infty}{\dfrac{ie^{i\omega t}}{\alpha\left(\omega-\omega_0\right)\left(\omega-\omega_1\right)\left(\omega-\omega_2\right)}}d\omega \space+\\
\\
&\lim_{r \to 0}{\left[ -\dfrac{e^{i\omega_0 t}}{\alpha}\int_{-\pi}^{0}{\dfrac{e^{itre^{i\omega}}}{\left(\omega_0+re^{i\omega}-\omega_1\right)\left(\omega_0+re^{i\omega}-\omega_2\right)}}d\omega \right]}+\\
\\
&\lim_{R \to \infty}{\left[\int_{0}^{-\pi}{\dfrac{ie^{itRe^{i\omega}}iRe^{i\omega}}{\alpha\left(Re^{i\omega}-\omega_0\right)\left(Re^{i\omega}-\omega_1\right)\left(Re^{i\omega}-\omega_2\right)}}d\omega \right]}\\
\\
&= P.V. \int_{-\infty}^{\infty}{\dfrac{ie^{i\omega t}}{\alpha\left(\omega-\omega_0\right)\left(\omega-\omega_1\right)\left(\omega-\omega_2\right)}}d\omega \space-\dfrac{\pi e^{i\omega_0 t}}{\alpha\left(\omega_0-\omega_1\right)\left(\omega_0-\omega_2\right)}\space + 0\\
\end{align*}$$
So for $t<0$
$$\begin{align*}&P.V. \int_{-\infty}^{\infty}{\dfrac{ie^{i\omega t}}{\alpha\left(\omega-\omega_0\right)\left(\omega-\omega_1\right)\left(\omega-\omega_2\right)}}d\omega \\
\\
&= \dfrac{\pi}{\alpha}\dfrac{ e^{i\omega_0 t}}{\left(\omega_0-\omega_1\right)\left(\omega_0-\omega_2\right)} + \dfrac{2\pi}{\alpha}\sum_{\omega_k \in \mathrm{LHP}} {\underset{z=\omega_k}{\mathrm{Res}}\left[\dfrac{e^{iz t}}{\left(z-\omega_0\right)\left(z-\omega_1\right)\left(z-\omega_2\right)}\right] }\\
\end{align*}$$
And, for example, for $t<0$, if both $\omega_1$ and $\omega_2$ are in the lower half plane, then working out the residues, one gets
$$\begin{align*}&P.V. \int_{-\infty}^{\infty}{\dfrac{ie^{i\omega t}}{\alpha\left(\omega-\omega_0\right)\left(\omega-\omega_1\right)\left(\omega-\omega_2\right)}}d\omega \\
\\
&= \dfrac{\pi}{\alpha}\dfrac{ e^{i\omega_0 t}}{\left(\omega_0-\omega_1\right)\left(\omega_0-\omega_2\right)} + \dfrac{2\pi}{\alpha}\dfrac{ e^{i\omega_1 t}}{\left(\omega_1-\omega_0\right)\left(\omega_1-\omega_2\right)} + \dfrac{2\pi}{\alpha}\dfrac{ e^{i\omega_2 t}}{\left(\omega_2-\omega_0\right)\left(\omega_2-\omega_1\right)}\\
\end{align*}$$
Update 2
For $t=0$, we'll need to use the following result, which I won't prove here. But it can be proven using $u$ substitution, the definition of Cauchy Principal Value, and two different contour integrations that have closed contours that are infinitely long rectangles, with one edge along the ral axis with a infintely small semi-circular excursion around the pole on the real axis:
$$ P.V. \int_{-\infty}^\infty{\dfrac{1}{x-z_k}}dx = \pi i \space\mathrm{sgn}\left(\Im\left[z_k\right]\right)$$
We'll attack the $t=0$ case by performing Partial Fraction Expansion of the integrand and then applying the above result to resultant three simpler integrals
$$\begin{align*} & P.V. \int_{-\infty}^{\infty}{\dfrac{ie^{i\omega 0}}{\alpha\left(\omega-\omega_0\right)\left(\omega-\omega_1\right)\left(\omega-\omega_2\right)}}d\omega \\
\\
&= P.V. \int_{-\infty}^{\infty}{\dfrac{i}{\alpha\left(\omega-\omega_0\right)\left(\omega-\omega_1\right)\left(\omega-\omega_2\right)}}d\omega \\
\\
&= P.V. \dfrac{i}{\alpha} \int_{-\infty}^\infty {\left[\dfrac{1}{\left(\omega_0-\omega_1\right)\left(\omega_0-\omega_2\right)\left(\omega-\omega_0\right)} \\
\\
+ \dfrac{1}{\left(\omega_1-\omega_0\right)\left(\omega_1-\omega_2\right)\left(\omega-\omega_1\right)} \\
\\
+ \dfrac{1}{\left(\omega_2-\omega_0\right)\left(\omega_2-\omega_1\right)\left(\omega-\omega_2\right)}\right]}d\omega\\
\\
&= \pi i\dfrac{i}{\alpha}\left[\dfrac{1}{\left(\omega_0-\omega_1\right)\left(\omega_0-\omega_2\right)}\mathrm{sgn}\left(\Im\left[\omega_0\right]\right)\\
\\
+ \dfrac{1}{\left(\omega_1-\omega_0\right)\left(\omega_1-\omega_2\right)}\mathrm{sgn}\left(\Im\left[\omega_1\right]\right)\\
\\
+ \dfrac{1}{\left(\omega_2-\omega_0\right)\left(\omega_2-\omega_1\right)}\mathrm{sgn}\left(\Im\left[\omega_2\right]\right)\right]\\
\end{align*}$$
\
Collecting all of the above results, we have
$$\begin{align*}&P.V. \int_{-\infty}^{\infty}{\dfrac{ie^{i\omega t}}{\alpha\left(\omega-\omega_0\right)\left(\omega-\omega_1\right)\left(\omega-\omega_2\right)}}d\omega \\
\\
&= 2\pi i\dfrac{i}{\alpha}\left[\dfrac{ e^{i\omega_0 t}}{\left(\omega_0-\omega_1\right)\left(\omega_0-\omega_2\right)}\dfrac{1}{2}\left[\mathrm{sgn}(t)+\mathrm{sgn}\left(\Im[\omega_0]\right)\right] + \\
\\
\dfrac{ e^{i\omega_1 t}}{\left(\omega_1-\omega_0\right)\left(\omega_1-\omega_2\right)} \dfrac{1}{2}\left[\mathrm{sgn}(t)+\mathrm{sgn}\left(\Im[\omega_1]\right)\right]+ \\
\\
\dfrac{ e^{i\omega_2 t}}{\left(\omega_2-\omega_0\right)\left(\omega_2-\omega_1\right)}\dfrac{1}{2}\left[\mathrm{sgn}(t)+\mathrm{sgn}\left(\Im[\omega_2]\right)\right]\right]\\
\end{align*}$$
Best Answer
The limit $R \rightarrow \infty$ is irrelevant.
Inside the integral, $r$ is not the variable of integration nor is it in the integration bounds, so under certain conditions, the limit of $r \rightarrow 0$ and the integration operation can be exchanged.
So can you try making the substitutions
$$r = \dfrac{1}{n}$$ $$f_n(\omega) = \frac{e^{i\frac{1}{n}e^{i\omega}t}}{\left(\frac{e^{i\omega}}{n}+\omega_0-\omega_1\right)\left( \frac{e^{i\omega}}{n}+\omega_0 -\omega_2\right)}$$ $$f(\omega) = \frac{1}{\left(\omega_0 -\omega_1\right)\left(\omega_0 -\omega_2\right)}$$ $$|f_n(\omega)| = \frac{1}{\left|\frac{e^{i\omega}}{n}+\omega_0-\omega_1\right|\left| \frac{e^{i\omega}}{n}+\omega_0 -\omega_2\right|}$$
assuming
$$n > \dfrac{1}{\min\left(|\omega_0-\omega_1|,|\omega_0-\omega_2| \right)}$$
and try applying the Dominated Convergence Theorem?
Namely, find an integrable function $g(\omega)$ such that
$$|f_n(\omega)| \le g(\omega)$$
for all finite real $\omega_0$ and all finite, distinct, complex $\omega_1$, $\omega_2$. (For any specific choice of distinct $\omega_0$, $\omega_1$, and $\omega_2$, it should be straightforward to select a suitable $g(\omega)$.)
I didn't, and still haven't, done this rigorously myself.
I'm thinking the constant function
$$g(\omega) = \dfrac{1}{\left[\min\left(|\omega_0-\omega_1|,|\omega_0-\omega_2| \right) - \dfrac{1}{n_{min}}\right]^2}$$
might work.
Update
In my original answer, for "Term 3", instead of parameterizing the small semi-circular contour around $\omega_0$ and going through the above gyrations, I could have left "Term 3" as
$$\lim_{r \to 0} \int_{C_r}{\dfrac{ie^{iz t}}{\alpha\left(z-\omega_0\right)\left(z-\omega_1\right)\left(z-\omega_2\right)}}dz$$
and then used the lemma in this answer.
Knowing that the clockwise orientation of the contour simply changes the sign of the answer, one can obtain
$$\lim_{r \to 0} \int_{C_r}{\dfrac{ie^{iz t}}{\alpha\left(z-\omega_0\right)\left(z-\omega_1\right)\left(z-\omega_2\right)}} dz = {\dfrac{\pi e^{i\omega_0 t}}{\alpha\left(\omega_0-\omega_1\right)\left(\omega_0-\omega_2\right)}}$$
right away.