= Solution
The corrected PDF integrand is
$$
f(\tau)\overline{G_k(\tau)}y^k\frac{dx\,dy}{y^2},
$$
so the integral is a <Petersson inner product>. On compact subsets it is harmless, while at the cusp the exponential decay of $f$ dominates the polynomial growth of the Eisenstein series; therefore it converges absolutely.
Decompose each nonzero pair uniquely into a positive common divisor times a primitive pair. For even $k$ this writes $G_k$ as $2\zeta(k)$ times the Eisenstein sum over $\Gamma_\infty\backslash SL_2(\mathbb Z)$. Unfolding the fundamental domain gives a constant multiple of
$$
\int_0^\infty\int_0^1 f(x+iy)y^{k-2}\,dx\,dy.
$$
The inner integral is the constant Fourier coefficient of the cusp form and is zero. Thus the original integral is zero, expressing the <orthogonality of cusp forms and holomorphic Eisenstein series>.
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