= Solution
Split $\phi=\phi_<+\phi_>$ and write $I_n=\int_{\Lambda/b<|k|<\Lambda}\frac{d^dk}{(2\pi)^d}G_0(k)^n$. The first <cumulant expansion>[cumulant] of $g_0\phi^4$ contains $6g_0\phi_<^2\langle\phi_>^2\rangle=6g_0I_1\phi_<^2$. The connected second cumulant of the two terms $3\lambda_0\phi_<\phi_>^2$ uses <Wick theorem> to give $\langle\phi_>^2(x)\phi_>^2(y)\rangle_c=2G_>(x-y)^2$. Keeping its leading local term, the quadratic coefficient after integrating out the shell is therefore
$$
\boxed{\mu'^2=\mu_0^2+12g_0I_1-18\lambda_0^2I_2
+O(g_0^2,g_0\lambda_0^2,\lambda_0^4)}.
$$
The quartic tadpole raises the mass parameter, while the pair of cubic vertices gives the negative bubble contribution. A cubic interaction also generates a linear tadpole $3\lambda_0I_1\phi_<$, which may be removed by shifting the background but is not part of $\mu'^2$.
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