= Solution
A <momentum-shell renormalization group> step has three parts. First split the field into slow and fast <Fourier modes>, $\phi=\phi_<+\phi_>$, and integrate over the shell $\Lambda/\zeta<|q|<\Lambda$. Second rescale $q'=\zeta q$, or equivalently $x'=x/\zeta$, to restore the <ultraviolet cutoff> to $\Lambda$. Third rescale the field so that the coefficient of $(\nabla\phi)^2/2$ returns to its chosen normalization. The resulting <free energy> has the same operator expansion but new couplings; iteration traces a <renormalization-group flow> in coupling space.
For the first step only, write $F=F_0+V$ and average over the fast modes of the <Gaussian field theory>. The <cumulant expansion> gives
$$
F_{\rm eff}[\phi_<]=F_0[\phi_<]+\langle V\rangle_>
-\frac12\langle V^2\rangle_{>,c}
+\frac16\langle V^3\rangle_{>,c}+\cdots.
$$
Here
$$
V=\lambda_0\int d^dx\,
(\phi_<^3+3\phi_<^2\phi_>+3\phi_<\phi_>^2+\phi_>^3).
$$
At order $\lambda_0^2$, the connected contraction of two $3\lambda_0\phi_<\phi_>^2$ vertices gives the low-momentum two-point term. Since $\langle\phi_>^2(x)\phi_>^2(y)\rangle_c=2G_>(x-y)^2$,
$$
\delta F^{(2)}=-9\lambda_0^2
\int d^dx\,d^dy\,\phi_<(x)G_>(x-y)^2\phi_<(y).
$$
Expanding at small external momentum and matching $(\mu'^2/2)\int\phi_<^2$ yields
$$
\boxed{\mu'^2=\mu_0^2-18\lambda_0^2
\int_{\Lambda/\zeta<|q|<\Lambda}\frac{d^dq}{(2\pi)^d}
\frac1{(q^2+\mu_0^2)^2}+O(\lambda_0^4).}
$$
The first cumulant also produces a term linear in $\phi_<$; it is removed by fixing the one-point function, or equivalently by a <field redefinition>, and does not change the displayed <one-particle-irreducible correlation function> correction to the mass.
The leading vertex correction is order $\lambda_0^3$. Taking $3\lambda_0\phi_<\phi_>^2$ from each of three vertices, the connected <Wick contractions> form a triangle. There are eight contractions, so the third cumulant contributes $27\times8/3!=36$ times the triangle integral. At zero external momentum,
$$
\boxed{\lambda'=\lambda_0+36\lambda_0^3
\int_{\Lambda/\zeta<|q|<\Lambda}\frac{d^dq}{(2\pi)^d}
\frac1{(q^2+\mu_0^2)^3}+O(\lambda_0^5).}
$$
For nonzero external momenta the three propagators carry the corresponding shifted loop momenta, with every internal line restricted to the fast shell.
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