= Solution
Use a dimensionless <Hamiltonian> in the functional weight $e^{-H}$, assume $\alpha>0$, and let $\phi(\mathbf x)=\int_{|\mathbf p|\leq\Lambda}d^Dp\,(2\pi)^{-D}e^{i\mathbf p\cdot\mathbf x}\widetilde\phi(\mathbf p)$. Reality means $\widetilde\phi(-\mathbf p)=\widetilde\phi(\mathbf p)^*$. At zero field, the <quadratic form> is
$$
H_0=\frac12\int_{|\mathbf p|\leq\Lambda}\frac{d^Dp}{(2\pi)^D}\left(\alpha^{-1}p^2+r_0\right)|\widetilde\phi(\mathbf p)|^2.
$$
In the <momentum-shell renormalization group>, split into $\phi_<+\phi_>$ below and above $\Lambda/b$. Disjoint Fourier supports make the <quadratic form> split into $H_0[\phi_<]+H_0[\phi_>]$. Integration over the shell gives a source-independent <Gaussian functional integral> multiplying the <partition function>, or an additive constant in the effective <free energy>; it leaves the slow-mode quadratic coefficients unchanged before rescaling. A uniform field has support only at zero momentum, so it does not change this shell integration.
Restore the <ultraviolet cutoff> by $\mathbf p'=b\mathbf p$, $\mathbf x'=\mathbf x/b$, and choose
$$
\phi'(\mathbf x')=b^{(D-2)/2}\phi_<(b\mathbf x'),\qquad \widetilde\phi'(\mathbf p')=b^{-(D+2)/2}\widetilde\phi_< (\mathbf p'/b).
$$
The measure, two gradients and two fields have scale factors $b^D$, $b^{-2}$ and $b^{-(D-2)}$, whose product is one. The mass term has factor $b^D b^{-(D-2)}=b^2$, while the uniform source term has factor $b^D b^{-(D-2)/2}=b^{(D+2)/2}$. Hence the <Gaussian momentum-shell scaling> is
$$
\boxed{\alpha^{-1}\longmapsto\alpha^{-1},\qquad r_0\longmapsto b^2r_0,\qquad h\longmapsto b^{(D+2)/2}h.}
$$
For a slowly varying nonuniform source the corresponding formula is $h'(\mathbf x')=b^{(D+2)/2}h(b\mathbf x')$. Here the nonuniform source on the right is its projection onto the retained <Fourier modes>; an eliminated source component contributes only to the field-independent Gaussian normalization. The field <engineering dimension> is $(D-2)/2$ and its Gaussian <anomalous dimension> is zero. The mass and uniform source are relevant perturbations of the <Gaussian fixed point>. Positivity of $\alpha$ is necessary for a stable <kinetic term> near this point. One can use a finite volume and positive mass as infrared regulators and then take the critical limit; at exactly zero mass the integral over the zero <Fourier mode> alone is not a normalized finite-volume <Gaussian measure>.
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