= Solution
The leading decay is into one complex particle and its antiparticle, which are distinguishable. Using $|\mathcal A|^2=g^2$ in <6/a> gives
$$
\boxed{\Gamma(\phi\to\varphi\bar\varphi)=\frac{g^2}{16\pi m}.}
$$
In four space-time dimensions a canonically normalized scalar has mass dimension one, so $g$ has mass dimension one. The amplitude starts at order $g$, a rate is quadratic in that amplitude, and the only mass scale in the massless-daughter decay is $m$. Those facts already require $\Gamma\propto g^2/m$, leaving only a dimensionless numerical factor to calculate.
For <low-energy charged-scalar scattering through a neutral scalar>, label the incoming particle and antiparticle as $1,2$ and their outgoing counterparts as $3,4$. Charge flow allows $s$-channel annihilation and $t=(p_1-p_3)^2$ exchange of the real massive scalar. It forbids a separate $u$-channel graph for this fixed assignment of distinguishable final charges. The tree amplitude is
$$
\boxed{\mathcal M=-g^2\left(\frac1{s-m^2}+\frac1{t-m^2}\right)
=\frac{2g^2}{m^2}+O\!\left(\frac{g^2s}{m^4},\frac{g^2t}{m^4}\right).}
$$
At low energy it is approximately angle independent. The massless two-body scattering formula then gives
$$
\boxed{\sigma_{\rm tot}\simeq\frac{|\mathcal M|^2}{16\pi s}
=\frac{g^4}{4\pi m^4s}.}
$$
In particular the requested scale estimate is $\sigma\sim g^4/(m^4s)$, not $g^4/m^6$ with the incoming-energy dependence omitted. Integrating out $\phi$ gives $\mathcal L_{\rm eff}=g^2(\varphi^\dagger\varphi)^2/(2m^2)$ at this order, whose complex-scalar four-leg combinatorics reproduce the same $2g^2/m^2$ amplitude. These are weak-coupling perturbative results about the specified vacuum; possible ultraviolet stabilizing interactions are not determined by the cubic term alone.
\Image[/past-exam-of-the-theoretical-physics-major-option-course-of-the-university-of-oxford/2015-complex-scalar-particle-antiparticle-scattering.png]
{title=The two allowed leading particle-antiparticle scattering diagrams}
{description=Original s-channel annihilation and t-channel exchange graphs. The internal dashed line is the massive neutral scalar; solid arrows show conserved complex charge. A u-channel graph would connect the wrong final charges for the fixed labeling.}
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