= Solution
After integration by parts, the quadratic ghost operator is $c_1(-\partial^2)+c_2M^2$. Matching its propagator to the scalar propagator after $M\mapsto m$ requires
$$
c_1=c_2=1.
$$
Differentiating $c_3\lambda\bar\eta\phi^2\eta$ with respect to its two identical scalar fields gives the ghost-scalar vertex $-2c_3\lambda$. Matching it to the four-scalar vertex therefore requires
$$
c_3=\frac12.
$$
The additional rules are an oriented ghost propagator $(p^2+M^2)^{-1}$, a two-scalar two-ghost vertex $-\lambda$, and a minus sign for every closed loop of <Grassmann-valued fields>.
Besides the scalar bubbles from part a, each channel now has a closed heavy-ghost bubble. Its two directed internal lines cannot be interchanged, so it has no scalar bubble's factor $1/2$. Consequently
$$
V_b^{(4)}=-\lambda_b+\lambda_b^2
\sum_{P\in\{p_1+p_2,p_1+p_3,p_1+p_4\}}
\left[\frac12I(P;m,\Lambda)-I(P;M,\Lambda)\right]
+O(\lambda_b^3).
$$
If $I_{M,\mathrm{os}}$ denotes the corresponding sum of heavy integrals, on-shell matching gives
$$
\boxed{\lambda_b=\lambda_{\mathrm{phys}}
+\lambda_{\mathrm{phys}}^2
\left(\frac12I_{m,\mathrm{os}}-I_{M,\mathrm{os}}\right)
+O(\lambda_{\mathrm{phys}}^3)}.
$$
The root continuously connected to $\lambda_b=\lambda_{\mathrm{phys}}$ is the perturbative, small positive root.
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