I am not sure whether this addresses yuor question, but one of the reasons for IR divergences in YM theory is the presence of bound states. The concept of bound states makes sense only on an infinite lattice, hence all problems associated with it arise only in that limit.
Bound states completely alter the renormalized action, since to get the S-matrix correct one has to add for each bound state a field with unrenormalized coefficent Z=0. (Cf. the somewhat cryptic remark in the middle of p.110 of Weinberg's Vol. 1.) Failure to do so results in severe divergences. Already for nonrelativistic scattering problems of a single particle in an external field, the Born series diverges.
In case of QED, one has therefore to treat the Coloumb external field problem in a completely different way than the standard case. I haven't seen anywhere a sensible treatment of the quantized field case. But see Chapter 6.2 of Derezinski's lecture notes for the case of a Dirac Fermion in an external electromagnetic field (i.e., QED with external field but without radiative corrections).
I don't know how this would show up in a lattice version of the theory.
I just found the following articles; haven't read them yet:
Baldicchi, M., and G. M. Prosperi. "Infrared behavior of the running coupling constant and bound states in QCD." Physical Review D 66.7 (2002): 074008.
Ganbold, Gurjav. "Hadron spectrum and infrared-finite behavior of QCD running coupling." Physics of Particles and Nuclei 43.1 (2012): 79-105.