The Dirac equation is more restrictive than the Klein-Gordon equation. For every solution to the Dirac equation, its components will be a solution of the Klein-Gordon equation, but the converse isn't true: if you form a spinor whose components are solutions of the Klein-Gordon equation, it might not solve the Dirac equation.
If we start with the Klein-Gordon equation for the whole spinor $\psi$ $$(\partial^2 +m^2)\psi =0 $$ the solution is $$\psi = u(\vec{p}) e^{-ipx} \quad \text{or} \quad \psi = v(\vec{p}) e^{+ipx}$$
where $u$ ans $v$ are arbitrary spinors. But, what happens if we plug these solutions in the original Dirac equation?
$$(\partial^2 + m^2)u(\vec{p})e^{-ipx} = -e^{-ipx}(\gamma^\mu p_\mu - m)u(\vec{p}) = 0$$
$$(\partial^2 + m^2)v(\vec{p})e^{+ipx} = -e^{+ipx}(\gamma^\mu p_\mu + m)v(\vec{p}) = 0$$
$u$ and $v$ are not arbitrary anymore! Instead, they must obey the stronger restriction $$(\gamma^\mu p_\mu - m)u(\vec{p}) = 0 \qquad\qquad (\gamma^\mu p_\mu + m)v(\vec{p}) = 0$$
If you consider only the Klein-Gordon equation, you're introducing extra "solutions" that don't really solve the Dirac equation.
Why does this happen? You can regard the Klein-Gordon equation as the "squared" version of the Dirac equation. And when you square an equation, you always get this nasty false solutions: if you have e.g the equation $(x-3)=5$ the solution is $x=8$, but if you square it $(x-3)^2 = 5^2$ then you have two solutions, $x=8$ and $x=-2$. The first equation implies the second, but the converse isn't true.
The field and the wavefunction look similar, but they don't really have much to do with each other. The main point of the field is to group the creation and annihilation operators in a convenient way, which we can use to construct observables. As usual I will start with the free theory.
If we want to find a connection to non-relativistic QM, the field equation is not the way to go. Rather, we should look at the states and the Hamiltonian, which are the basic ingredients of the Schrödinger equation. Let's look at the Hamiltonian first. The usual procedure is to start with the Lagrangian for the free scalar field, pass to the Hamiltonian, write the field in terms of $a$ and $a^\dagger$, and plug that into $H$. I will assume you know all this (it's done in every chapter on second quantization in every QFT book), and just use the result:
$$H = \int \frac{d^3 p}{(2\pi)^3}\, \omega_p\, a^\dagger_p a_p$$
where $\omega_p = \sqrt{p^2+m^2}$. There's also a momentum operator $P_i$, which turns out to be
$$P_i = \int \frac{d^3 p}{(2\pi)^3}\, p_i\, a^\dagger_p a_p$$
Using the commutation relations it is straightforward to calculate the square of the momentum, which we will need later:
$$P^2 = P_i P_i = \int \frac{d^3 p}{(2\pi)^3}\, p^2\, a^\dagger_p a_p + \text{something}$$
where $\text{something}$ gives zero when applied to one particle states, because it has two annihilation operators next to each other.
Now let's see how to take the non-relativistic limit. We will assume that we are dealing only with one-particle states. (I don't know how much loss of generality this is; the free theory doesn't change particle number so it shouldn't a big deal, and also we usually assume a fixed number of particles in regular QM.) Let's say that in the Schrödinger picture we have a state that at some point is written as $|\psi\rangle = \int \frac{d^3 k}{(2\pi)^3} f(k) |k\rangle$, where $|k\rangle$ is a state with three-momentum $\mathbf{k}$. $f(k)$ should be nonzero only for $k \ll m$. Now look what happens if we apply the Hamiltonian. Since we only have low momentum, over the range of integration we can approximate $\omega_p$ as $m+p^2/2m$ and ignore the constant rest energy $m$.
$$H|\psi\rangle = \int \frac{d^3p}{(2\pi)^3} \frac{p^2}{2m} a^\dagger_p a_p \int \frac{d^3 k}{(2\pi)^3} f(k) |k\rangle \\
= \int \frac{d^3p}{(2\pi)^3} \frac{d^3 k}{(2\pi)^3} \frac{p^2}{2m} f(k) a^\dagger_p a_p |k\rangle \\
= \int \frac{d^3p}{(2\pi)^3} \frac{d^3 k}{(2\pi)^3} \frac{p^2}{2m} f(k) (2\pi)^3 \delta(p-k) |k\rangle \\
= \int \frac{d^3 k}{(2\pi)^3} \frac{k^2}{2m} f(k) |k\rangle = \frac{P^2}{2m} |\psi\rangle
$$
So if $|\psi\rangle$ is any one-particle state (which it is because the states of definite momentum form a basis), we have that $H|\psi\rangle = P^2/2m |\psi\rangle$. In other words, on the space of one-particle states, $H = P^2/2m$. The Schrödinger equation is still valid in QFT, so we can immediately write
$$\frac{P^2}{2m} |\psi\rangle = i \frac{d}{dt} |\psi\rangle$$
This is the Schrödinger equation for a free, non-relativistic particle. You will notice that I kept some concepts from QFT, particularly the creation and annihilation operators. You can do this no problem, but working with $a$ and $a^\dagger$ in QM is not particularly useful because they create and destroy particles, and we have assumed the energy is not high enough to do that.
Handling interactions is more complicated, and I fully admit I'm not sure how to include them here in a natural way. I think part of the issue is that interactions in QFT are quite limited in their form. We would have to start with the full QED Lagrangian, remove the $F_{\mu\nu}F^{\mu\nu}$ term since we aren't interested in the dynamics of the EM field itself, maybe set $A_i = 0$ if we don't care about magnetic fields, and see what happens to the Hamiltonian. Right now I'm not up to the task.
I hope I can convinced you that this newfangled formalism reduces to QM in a meaningful way. A noteworthy message is that the fields themselves don't carry a lot of physical meaning; they're just convenient tools to set up the states we want and calculate correlation functions. I learned this from reading Weinberg; if you're interested in these kinds of questions, I recommend you do so too after you've become more comfortable with QFT.
Best Answer
In the center of mass frame, let $p_1$ be the inbound photon, $p_2$ the inbound electron, $p_3$ the scattered photon, $p_4$ the scattered electron.
\begin{equation*} p_1=\begin{pmatrix}\omega\\0\\0\\ \omega\end{pmatrix} \qquad p_2=\begin{pmatrix}E\\0\\0\\-\omega\end{pmatrix} \qquad p_3=\begin{pmatrix} \omega\\ \omega\sin\theta\cos\phi\\ \omega\sin\theta\sin\phi\\ \omega\cos\theta \end{pmatrix} \qquad p_4=\begin{pmatrix} E\\ -\omega\sin\theta\cos\phi\\ -\omega\sin\theta\sin\phi\\ -\omega\cos\theta \end{pmatrix} \end{equation*}
where $E=\sqrt{\omega^2+m^2}$.
It is easy to show that
\begin{equation} \langle|\mathcal{M}|^2\rangle = \frac{e^4}{4} \left( \frac{f_{11}}{(s-m^2)^2} +\frac{f_{12}}{(s-m^2)(u-m^2)} +\frac{f_{12}^*}{(s-m^2)(u-m^2)} +\frac{f_{22}}{(u-m^2)^2} \right) \end{equation}
where
\begin{equation} \begin{aligned} f_{11}&=-8 s u + 24 s m^2 + 8 u m^2 + 8 m^4 \\ f_{12}&=8 s m^2 + 8 u m^2 + 16 m^4 \\ f_{22}&=-8 s u + 8 s m^2 + 24 u m^2 + 8 m^4 \end{aligned} \end{equation}
for the Mandelstam variables $s=(p_1+p_2)^2$, $t=(p_1-p_3)^2$, $u=(p_1-p_4)^2$.
Next, apply a Lorentz boost to go from the center of mass frame to the lab frame in which the electron is at rest.
\begin{equation*} \Lambda= \begin{pmatrix} E/m & 0 & 0 & \omega/m\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ \omega/m & 0 & 0 & E/m \end{pmatrix}, \qquad \Lambda p_2=\begin{pmatrix}m \\ 0 \\ 0 \\ 0\end{pmatrix} \end{equation*}
The Mandelstam variables are invariant under a boost. \begin{equation} \begin{aligned} s&=(p_1+p_2)^2=(\Lambda p_1+\Lambda p_2)^2 \\ t&=(p_1-p_3)^2=(\Lambda p_1-\Lambda p_3)^2 \\ u&=(p_1-p_4)^2=(\Lambda p_1-\Lambda p_4)^2 \end{aligned} \end{equation}
In the lab frame, let $\omega_L$ be the angular frequency of the incident photon and let $\omega_L'$ be the angular frequency of the scattered photon. \begin{equation} \begin{aligned} \omega_L&=\Lambda p_1\cdot(1,0,0,0)=\frac{\omega^2}{m}+\frac{\omega E}{m} \\ \omega_L'&=\Lambda p_3\cdot(1,0,0,0)=\frac{\omega^2\cos\theta}{m}+\frac{\omega E}{m} \end{aligned} \end{equation}
It follows that \begin{equation} \begin{aligned} s&=(p_1+p_2)^2=2m\omega_L+m^2 \\ t&=(p_1-p_3)^2=2m(\omega_L' - \omega_L) \\ u&=(p_1-p_4)^2=-2 m \omega_L' + m^2 \end{aligned} \end{equation}
Compute $\langle|\mathcal{M}|^2\rangle$ from $s$, $t$, and $u$ that involve $\omega_L$ and $\omega_L'$. \begin{equation*} \langle|\mathcal{M}|^2\rangle= 2e^4\left( \frac{\omega_L}{\omega_L'}+\frac{\omega_L'}{\omega_L} +\left(\frac{m}{\omega_L}-\frac{m}{\omega_L'}+1\right)^2-1 \right) \end{equation*}
From the Compton formula \begin{equation*} \frac{1}{\omega_L'}-\frac{1}{\omega_L}=\frac{1-\cos\theta_L}{m} \end{equation*}
we have \begin{equation*} \cos\theta_L=\frac{m}{\omega_L}-\frac{m}{\omega_L'}+1 \end{equation*}
Hence \begin{equation*} \langle|\mathcal{M}|^2\rangle= 2e^4\left( \frac{\omega_L}{\omega_L'}+\frac{\omega_L'}{\omega_L}+\cos^2\theta_L-1 \right) \end{equation*}
The differential cross section for Compton scattering is \begin{equation*} \frac{d\sigma}{d\Omega}\propto \left(\frac{\omega_L'}{\omega_L}\right)^2\langle|\mathcal{M}|^2\rangle \end{equation*}