= Solution
It is useful to regulate the number of modes first, so the <functional integral> identities reduce to ordinary <integration by parts>. Let $C$ be the Gaussian covariance, $F=C^{-1}$, $W=e^{-S_1}$ and put a dot for $\Lambda\partial_\Lambda$. The matrix identity $\dot F=-F\dot C F$ gives
$$
\dot e^{-S_0}=\frac12(F\phi)^T\dot C(F\phi)e^{-S_0}.
$$
The second field derivative of the Gaussian is
$$
\partial_a\partial_b e^{-S_0}
=[(F\phi)_a(F\phi)_b-F_{ab}]e^{-S_0}.
$$
Hence, after two <integrations by parts>,
$$
\dot Z=\int d\phi\,e^{-S_0}\left[\dot W+\frac12\dot C_{ab}\partial_a\partial_bW\right]
+\frac12\operatorname{Tr}(F\dot C)Z.
$$
The imposed flow makes the bracket vanish. The remaining trace is independent of the fields and only changes the Gaussian normalization. Since the free Gaussian normalization is $\mathcal N=(\det(2\pi C))^{1/2}$, $\dot{\log\mathcal N}=\operatorname{Tr}(F\dot C)/2$. Therefore
$$
\boxed{\Lambda\partial_\Lambda(Z/\mathcal N)=0.}
$$
Equivalently, $Z$ is cutoff independent after discarding the stated overall rescaling. This is the <Gaussian covariance differentiation identity> behind the <Polchinski equation>.
With the Fourier convention above and functional derivatives satisfying $\delta\widetilde\phi(p)/\delta\widetilde\phi(q)=\delta^{(4)}(p-q)$, contraction with $\dot C$ becomes $\int d^4p\,(2\pi)^4\dot C_\Lambda(p)\delta^2/[\delta\widetilde\phi(p)\delta\widetilde\phi(-p)]$. Thus \b[the numerator $(2\pi)^4$ in the printed flow is consistent with this derivative convention]; it must not be changed independently of the convention.
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