= Solution
Let $G_5$ be the connected time-ordered five-point function. The <LSZ reduction formula> gives, up to one factor $Z^{-1/2}$ per external leg,
$$
\begin{aligned}
\langle k_1k_2k_3,\mathrm{out}|p_1p_2,\mathrm{in}\rangle_c
={}&\prod_{r=1}^3\left[i\int d^4x_r\,e^{ik_r\cdot x_r}(\Box_{x_r}+m^2)\right]\\
&\times\prod_{s=1}^2\left[i\int d^4y_s\,e^{-ip_s\cdot y_s}(\Box_{y_s}+m^2)\right]
G_5(x_1,x_2,x_3,y_1,y_2).
\end{aligned}
$$
Translation invariance factors this as
$$
i(2\pi)^4\delta^{(4)}\!\left(p_1+p_2-\sum_{r=1}^3k_r\right)\mathcal M_{2\to3}.
$$
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