"However, if you come up with a theory that makes the exact same predictions as QM and GR except for the BHILP, then your theory is not very interesting, since we will never be able to test this theory, and there are other inconsistencies between QM and GR that you haven't solved, so we can't just rest on our laurels and say physics is over."
I think this is wrong. There are non-empirical reason reasons we might prefer the new theory to the old one. For example, we know QM are GR are inconsistent, so if I give you a parsimonious, consistent theory that captures both QM and GR in the appropriate limits, then that's a great reason to prefer it. In principle, we could prove theorems (and some theorems of this form have been proved) that pin down the possible resolutions of the GR/QM inconsistency. If we tighten the net so that only one theory remains, then obviously we should choose to believe that theory.
Now, will that actually happen? I don't know. But merely waving your hands and going "not testable" is not enough, because we all accept that mathematical consistency ought to have a huge influence on theory choice. You need to make some argument about BHILP specifically saying that theoretical arguments will never produce productive physics.