"a universe in which the Earth is fixed, and all the laws of physics are changed to make that universe indistinguishable from our own," would be indistinguishable from our own"
Not quite what I was saying, really.
Remember, the "laws" of physics are not laws. They are human models, and they fray at the edges (faster-than-light universal expansion, for example). There is a uniformitarian assumption built into the models of physics which isn't verifiable, and which could be wrong.
If you assume that the physics models are laws, observed and concrete and Right (capital R), then sure, you'd have to really change the "laws" of physics to make an earth-centered universe look exactly as the one we have now.
But you can't really go that far. Because the "laws" of physics aren't laws at all, they're models, and it could be that they're wrong. It could be that the earth is, in fact, the center of the real universe, and that our laws of physics are wrong, and do not in fact represent reality, that the REAL laws of physics - which we don't understand, and which are quite contrary to our erroneous models - cause the universe to behave exactly as we see it behave, given that our earth is at the very center of everything and does not move.
You could model the universe that way. "Problems", such as the retrograde motion of the planets, could be expressed mathematically. What would be absent would bt eh neat and simple explanation of our current model, that the suns' gravity causes the planets to revolve elliptically around it. Instead, we would observe (as we DO observe, if we choose to do so) that the planets sun and universe revolve around the earth, but that the planets do not do so in elliptical orbits. Obviously, then the gravity model would be wrong and some other entity or juxtaposition of entities would be causing the planets to do that.
It would certainly be a messy way to model the universe, messier than the current model which has gravity as an explanation for the behavior. But it COULD be in the real, actual, universe as it really is, that gravity ISN'T the explanation, and that everything really DOES revolve around a fixed earth, exactly as we see it. This does not mean postulating a new universe. It's THIS universe. It means junking the "laws" of physics and starting over. What we'd end up with would be a proliferation of entities and of mysteries, as compared to what we think we know now. That doesn't mean it would be WRONG. It does mean that it's not very likely at all.
You might run into a bit of bother modelling the phases of Venus.
Attempt to add "The Lyman Alpha Forest, CMB, and GR to you model. Just ain't gonna happen. Geocentrism is dead. (except for certain models that make stellar navigation easier to compute, however, those are very specific and cannot be used in the general sense)