Two-electron Integrals over Gaussian Plane Waves
I'm not sure if anyone has done it, but the closest I can think of is the GPW (mixed Gaussian Plane Wave) method. I think that involves a mapping between Gaussian and PW basis. It's DFT, but just look for the Coulomb integrals.

General idea:

Lippert, Gerald, et al. "A hybrid Gaussian and plane wave density functional scheme." Molecular Physics 92.3 (1997): 477-488.

The implementation can be found here:

VandeVondele, Joost, et al. "Quickstep: Fast and accurate density functional calculations using a mixed Gaussian and plane waves approach." Computer Physics Communications 167.2 (2005): 103-128.