= Solution
Split the Fourier modes into $|p|<\Lambda/b$ and the shell $\Lambda/b<|p|<\Lambda$. The source is uniform, so it couples only to the zero mode, which belongs to the retained part. The <Gaussian field theory> has no couplings between the shell and retained modes. Integrating out the shell therefore contributes only a source-independent determinant to the <free energy>, without changing the quadratic coefficients of the retained Hamiltonian.
Restore the cutoff by $x=bx'$ and choose the field normalization $\phi_<(bx')=b^{-(D-2)/2}\phi'(x')$. Counting the measure, derivative and field factors explicitly gives
$$
\int d^Dx\,\frac1{2\alpha}|\nabla\phi_<|^2
=\int d^Dx'\,\frac1{2\alpha}|\nabla'\phi'|^2,
$$
$$
\int d^Dx\,\frac{m^2}{2}\phi_<^2
=\int d^Dx'\,\frac{b^2m^2}{2}\phi'^2,\qquad
\int d^Dx\,h\phi_<
=\int d^Dx'\,b^{(D+2)/2}h\phi'.
$$
Thus in the convention that preserves the kinetic normalization, the <Gaussian momentum-shell scaling> is
$$
\boxed{\alpha'=\alpha,\qquad (m^2)'=b^2m^2,\qquad h'=b^{(D+2)/2}h.}
$$
The field has <engineering dimension> $(D-2)/2$ and zero <anomalous dimension>. The measure Jacobian and shell determinant supply the additive constant described in Q2. A different choice of field rescaling can redistribute factors into $\alpha$, but must give the same physical transformed <correlation length> $\xi'=\xi/b$.
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