It is not unreasonable that a massless field behaves in a way that is totally different from a massive one with arbitrarily small mass. Already at an elementary level, you can always perform a Lorentz transformation to the rest frame of a massive excitation; there is no such transformation for a massless one.

Whenever you encounter a mathematical ambiguity in a physics problem, it means that you have not taken into account all the necessary physics information. Physics has to tell you which order of limits is the relevant one. The $\epsilon $ prescriptions you are using usually serve to implement causality in the propagators you are evaluating - that may yield a clue.

Without knowing the full details of what you're calculating, one possibility is that you are considering the propagation of an actual massive particle, regardless of how small the mass is. In that case, $\epsilon $ has to be kept much smaller than $\alpha $, i.e., the order of limits is opposite to the case of the massless field. Another possibility is that you are treating the propagation of a massless particle, and merely introducing a mass as an infrared regulator at an intermediate stage. That is a rather subtle thing to do! One would then require that final physical results are not altered; e.g., that the additional polarization state induced for a photon is not counted in, say, a partition function.