= Solution
Rearrange the continuum equation as
$$
(\nabla^2+k^2)\psi_k=\frac{2mg}{\hbar^2}d_k\delta^{(3)}(\mathbf r).
$$
The incident <plane wave> contributes zero to the left side. The outgoing <Green function> identity gives $-4\pi f_k\delta^{(3)}(\mathbf r)$ for the scattered term, hence
$$
\boxed{f_k=-\frac{mg}{2\pi\hbar^2}\,d_k.}
$$
This fixes both the sign and the three-dimensional $4\pi$ factor. The model is an isotropic point coupling, so the scattered amplitude has no angular dependence. Take $g$ real, as required by the displayed Hermitian pair of coupling equations; an arbitrary phase of the discrete state can be chosen accordingly.
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