a proca field
I think a mass term for the electromagnetic field breaks the gauge invariance of the theory. You are following a route which is technically simpler, but obscures the essential fact that the 4-potential is a gauge field. Namely, you can always set the mass of the photons to zero in the end of all the calculations and obtain the same result without ever discussing gauge invariance. But, then, your coupling term, the very topic of this thread, is inserted 'by hand' and does not have an obvious meaning. Also, by the Ward identities, I think the photon cannot acquire mass in any order of perturbation theory.