1) OP is basically wondering how Weinberg on the middle of p. 112 can extend the integration region from$^1$
$${\cal J}^{\pm}_{\beta}~=~ \int_{m_{\alpha}}^{\infty} \!dE_{\alpha}\frac{e^{-iE_{\alpha}t}g(E_{\alpha})T_{\beta\alpha}^{\pm}} {E_{\alpha}-E_{\beta}\pm i0^{+}}$$
to include the negative real axis
$${\cal J}^{\pm}_{\beta}~=~ \int_{-\infty}^{\infty} \!dE_{\alpha}\frac{e^{-iE_{\alpha}t}g(E_{\alpha})T_{\beta\alpha}^{\pm}} {E_{\alpha}-E_{\beta}\pm i0^{+}},$$
where $g:E_{\alpha}\mapsto g(E_{\alpha})$ is a meromorphic function?
2) That Weinberg (implicitly) assumes meromorphicity of the $g:D\subseteq \mathbb{C}\to \mathbb{C}$ function can be deduced further down on p. 112, where he writes that
[...] we can close the contour of integration for the integration variable $E_{\alpha}$ [...],
which is a clear reference to the residue theorem, which in turn assumes meromorphicity. Also Weinberg writes on the same page$^1$
[...] The functions $g(E_{\alpha})$ and $T_{\beta\alpha}^{\pm}$ may, in general, be expected to have some singularities at values of $E_{\alpha}$
with finite [...] imaginary parts [...]
So there is little doubt that Weinberg assumes meromorphicity of $g$.
3) On the other hand, on the bottom of p. 109, Weinberg writes$^1$
[...]Therefore, we must consider wave-packets, superpositions $\int\! dE_{\alpha}~g(E_{\alpha})\Psi_{\alpha}$ of states, with an amplitude $g(E_{\alpha})$ that is non-zero and smoothly varying over some finite range $\Delta E$ of energies.[...]
Now according to the identity theorem for holomorphic functions, if a function $g:D\subseteq \mathbb{C}\to \mathbb{C}$ is zero on a subset $S\subseteq D$ that has an accumulation point $c$ in the domain $D$, then $g\equiv 0$ is identically zero. However, any interval $I\subseteq \mathbb{R}$ on the real line of non-zero length has accumulation points. So if Weinberg in above quote literally means that $g$ is mathematically zero outside some finite interval $I\subseteq \mathbb{R}$, then $g\equiv 0$ would be identically zero in the whole complex plane.
Of course Weinberg doesn't mean that. He just means that $g$ outside some finite range takes so small values, that to the precision $\epsilon$ that we are working, it doesn't matter whether we include the integration region $\mathbb{R}\backslash I$, or not.
In particular, mathematically speaking, Weinberg has only proven the condition
$$\tag{3.1.12} \int_{m_{\alpha}}^{\infty} \!dE_{\alpha}~
e^{-iE_{\alpha}t} g(E_{\alpha}) \Psi^{\pm}_{\alpha}~\longrightarrow~
\int_{m_{\alpha}}^{\infty} \!dE_{\alpha}~
e^{-iE_{\alpha}t} g(E_{\alpha}) \Phi_{\alpha}
~\text{for}~ t\to\mp\infty \qquad $$
within some precision $\epsilon$. However, the precision $\epsilon$ can be made arbitrarily fine by preparing more and more sharply defined wavepackets $g$.
4) If one would like to have a concrete example of a $g$ function, one may think of a Lorentzian function (aka. Breit–Wigner or Cauchy distribution),
$$g(E_{\alpha}) ~=~ \frac{1}{\pi}\frac{\delta}{(E_{\alpha}-E_0)^2+\delta^2}, \qquad \int_{-\infty}^{\infty}\! dE_{\alpha}~g(E_{\alpha})~=~1,$$
for appropriate choices of constants $E_0$ and $\delta$.
5) Finally, one should not loose sight of Weinberg's main goal in Section 3.1, namely to argue the $\pm i0^{+}$ prescription in the Lippmann-Schwinger equations
$$\tag{3.1.17}\Psi^{\pm}_{\alpha}
~=~\Phi_{\alpha}+\int\!d\beta\frac{T_{\beta\alpha}^{\pm}\Phi_{\beta}} {E_{\alpha}-E_{\beta}\pm i0^{+}}.$$
The Lippmann-Schwinger equations (3.1.17) are not an approximation, and they are independent of the choice of wavepacket $g$.
--
$^1$ To simplify the discussion, we have taken the liberty to replace Weinberg's more general $\alpha$-integration with just an $E_{\alpha}$-integration. Here
$$\tag{3.1.4} \int \!d\alpha \cdots \equiv \sum_{n_1\sigma_1n_2\sigma_2\cdots}\int d^3p_1 d^3p_2 \cdots$$
Changing integration variable from $E_{\alpha}$ to momenta does not solve OP's problem, essentially because we still have to pick the branch of the pertinent square root that has positive real part, so that it doesn't bring us any closer in understanding negative energies.
Best Answer
The idea here is, that you define in- and outgoing states that are eigenstates of the non-interacting theory. Only at a finite time in the past did the interaction start and at a finite time in the future, the interaction stops.
We know exactly how to deal with free states, and we want to use this knowledge even when we introduce interactions. By introducing free "in" and "out" states, we can separate the (exactly solvable) free theory from the interacting theory, whose solution can only be given by a perturbation series.