= Solution
Take $\phi(p_1)+\bar\phi(p_2)\to\phi(p_3)+\bar\phi(p_4)$ with the <Mandelstam variables> of <2/c/i>. The allowed <tree-level Feynman diagram> is the crossed $u$-channel $\Phi$ exchange in panel (c) of <1/b>'s figure. At one end the incoming particle and outgoing <antiparticle> transfer charge $+2$; the other end absorbs that charge. The zero-charge $s$- and $t$-channel assignments are forbidden by the <interaction vertices>. Thus
$$
\boxed{\mathcal M_{\phi\bar\phi}=-\frac{g^2}{u-M^2+i\epsilon}.}
$$
In the <center-of-mass frame> with $|\mathbf p|\ll m$, $s=4m^2+O(|\mathbf p|^2)$ and $u=O(|\mathbf p|^2)$. For $M\gg m$, both <scattering amplitudes> approach $g^2/M^2$. The elastic two-body <Lorentz-invariant phase space> and incident flux agree for the two experiments. Using
$$
\frac{d\sigma}{d\Omega}=\frac{|\mathcal M|^2}{64\pi^2s}
$$
for distinguishable final particles, and an extra factor $1/2!$ when integrating the identical-particle final state over the whole sphere, gives
$$
\sigma_{\phi\phi}\simeq\frac{g^4}{32\pi sM^4},\qquad
\sigma_{\phi\bar\phi}\simeq\frac{g^4}{16\pi sM^4},\qquad
\boxed{\frac{\sigma_{\phi\phi}}{\sigma_{\phi\bar\phi}}\longrightarrow\frac12.}
$$
There is no additional division for identical incoming particles in the definition of a beam <scattering cross-section>. Before taking $M/m\to\infty$, the zero-<momentum> ratio is $\tfrac12[M^2/(M^2-4m^2)]^2$, provided the threshold is away from the heavy pole.
Near $s=M^2$, the bare $s$-channel <scalar propagator> makes fixed-order <perturbation theory> fail: repeated <self-energy> insertions contain powers of $\Sigma_\Phi(s)/(s-M^2)$. The $\Phi\to\phi\phi$ channel is open for $M>2m$, giving the leading <decay width>
$$
\Gamma_\Phi=\frac{g^2}{32\pi M}\sqrt{1-\frac{4m^2}{M^2}}.
$$
Resumming the <self-energy> gives a finite <Breit-Wigner resonance cross section>, with denominator $s-M_{\rm pole}^2+iM_{\rm pole}\Gamma_\Phi$ near the pole, rather than an infinite <scattering cross-section>. Also, $s\simeq M^2$ with $M\gg m$ lies outside the assumed nonrelativistic incoming kinematics; the ratio $1/2$ is not a resonance prediction.
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