The parts of GR we trust in order to interpret the data from our instruments is trusted precisely because there is evidence to back up those parts of the theory. We have no idea if that theory holds on the other side of a boundary across which no causation can occur.
The data we have about black holes is nothing more than evidence of objects with a size and density which are only consistent with black holes according to GR. Without assuming GR is true, you have only evidence of unusually dense objects, nothing more.
The alignment of certain evidence of certain parts of GR does not necessarily imply the correctness of the parts of GR that are not relevant to the evidence we have access to.
