You tell me; Copenhagen is the one that says collapse exists. It sounds like maybe we are on the same page that collapse isn't necessary to explain quantum mechanical observations? In that case, we are both Everettians :).
In some cases it is a safe approximation to ignore those extra states for the remainder of our experiment/calculations, but with a small change to the experiment we can make that a bad approximation.
a) You do the measurement first on the "screen" side (and project the quantum state of the pair of photons according to the measurement, the "extra universes" disappear). You do then the measurements on the "idler" side (and project again the quantum state according to standard QM).
b) You modify slighly the setup to reverse the order of the measurements. You do the measurement first on the "idler" side (and project the quantum state of the pair of photons according to the measurement, the "extra universes" disappear). You do then the measurements on the "screen" side (and project again the quantum state according to standard QM).
QM predicts that the outcomes in the original experiment (a) and the "reversed" experiment (b) are the same. And those predictions are verified empirically.
I'm not sure what's the problem. Why do have to "ask" if the answer doesn't really matter? What answer more consistent than "it doesn't really matter" would you like? Anyway (standard) QM is a non-relativistic theory, QFT may be more satisfactory from that point of view.
Re: "interference tagging" - what I mean is that first you detect the photons and later check if they "did happen" to go through two slits (interference appears) or one slit (no interference). But the interference pattern is not visible for a single photon and at that point the individual events are still a superposition of both possibilities (so for the events at a certain position part of them will be in the end identified as coming trough one slit and some of them from both). Only after the second measurement is done you know how to group the previoulsy recorded events to see the interference. It's not that the later measurement causes interference to appear. Or at least it doesn't affect at all where the photons were detected, it just lets you know how to group the existing events to make it apparent (selecting only those where, once the full mesurement on the pair has been done, the path taken remains uncertain).
Edit: maybe this picture helps https://upload.wikimedia.org/wikipedia/commons/thumb/c/c8/De...
If all the events are taken together there is no interference pattern. But when they are grouped according to where the second photon is detected in two cases there is still no interference but in the other cases complementary interference patterns appear.