This sounds strange, given that in the Wilsonian approach it is quite clear what to do with this sort of infinity. The non-renormalizable theory has to be understood as an effective field theory, and there is a critical length at which the effective field theory fails. One can now assume that all the terms have a similar order of magnitude at this critical distance. And then one can see what survives in the large distance limit. These are, first of all, renormalizable components. Then there are a few lowest order non-renormalizable ones.When the gauge symmetry is explicitly broken the theory is unrenormalizable, which means that (unlike in the renormalizable case) the manifold of perturbatively (and canonically) renormalized field theories associated with it is infinite-dimensional, due to the infinitely many counterterms. Nothing at all is known about the resulting nonperturbative situation.
Given this purely qualitative consideration, one can already make a qualitative guess. The renormalizable part of a massive gauge theory is the massless, gauge-invariant theory. The massive part gives a short distance force, thus, vanishes for large distances anyway (as suggested above).
We have to do this anyway, given that gravity is non-renormalizable, and this gives already a nice suggestion for the critical length. So, all that remains to be done would be to consider the lowest order non-renormalizable terms and what they give, say, for the massive gauge theories at Planck scale.
If this has been done somewhere, I would be very interested in the results. It not, I would be interested to understand why this has not been done.
(The same question appears for gauge theories with anomalies. Here I would guess that one can even predict qualitatively with more certainty that the anomalous part will be suppressed, thus, becomes very weak in comparison with the non-anomalous gauge fields. Here I would wonder why this method has not been used to extend the SM gauge group, given that one would not even have to invent a mechanism to suppress the additional gauge fields.)