= Solution
For a <razor-thin disk>, integrate the delta function in height to get
$$
\Phi_\epsilon(\mathbf R)=-G\int\frac{\Sigma(\mathbf R')\,d^2R'}{\sqrt{|\mathbf R-\mathbf R'|^2+\epsilon^2}}.
$$
The corresponding horizontal force kernel is proportional to $(\mathbf R-\mathbf R')/(|\mathbf R-\mathbf R'|^2+\epsilon^2)^{3/2}$. It removes the short-distance point-force singularity and smooths forces on separations of order $\epsilon$. This is <height-evaluation gravitational softening>.
To represent finite vertical thickness, choose $\epsilon$ of order the disc <disk scale height> $H$. There is no universal exact numerical choice: a true vertical <mass density> profile produces the Fourier reduction factor $\Sigma^{-1}\int\rho(z)e^{-k|z|}dz$, which is not generally $e^{-k\epsilon}$. Matching the long-wave term gives $\epsilon=\langle|z|\rangle_\rho$, the <vertical-profile softening match>. For an exponential vertical profile this mean is $H$, while its full reduction factor is $(1+kH)^{-1}$.
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