Yes, but equivalence allows acceleration and gravity to cause motion in two different ways in the two differing views.
Gravity doesn’t cause motion in GR. That's where the disconnect is. In GR gravity isn’t a force and only a force can cause motion. FE wants to say, “In GR gravity isn’t a force, therefore in GR gravity doesn’t cause motion.” but GR doesn’t say that gravity causes motion. They are disputing a conclusion that GR doesn’t make.
Does FE state that the effect of acceleration is the warping of spacetime? I don't believe it does but admittedly don't know.
Tom I know rejects it, but I don’t think I’ve ever seen an “official position”. Logically, they have to reject it (so at least Tom is consistent in that) because if acceleration warps space time then the EP as justification for UA goes out the window (no pun intended). It allows for not just an alternate, but a better explanation for gravity than UA.
If you accept that acceleration warps spacetime and maintain consistency between Newtonian and GR concepts, this is where a coherent argument leads...acceleration causes spacetime warp, acceleration and gravity are the same thing, therefore, gravity is the warping of spacetime, not some force that accelerates the earth up.
IF the earth were flat and accelerating upwards due to UA the water and bottle would behave exactly the same way as under GR. Which one happens to be true is irrelevant to the test.
Since the EP doesn’t apply, there is no reason to assume that, and other reasons to assume that it would not be the case.