Hi again.
Well, you need more than just the Equivalence Principle (EP) to derive General Relativity. There are other theories of gravity that are consistent with the EP. The Brans-Dicke theory of gravity is one obvious example.
...the Brans–Dicke theory of gravitation (sometimes called the Jordan–Brans–Dicke theory) is a theoretical framework to explain gravitation. It is a competitor of Einstein's theory of general relativity...
...Both Brans–Dicke theory and general relativity are examples of a class of relativistic classical field theories of gravitation, called metric theories....
....All metric theories satisfy the Einstein equivalence principle, .....
[Taken from https://en.wikipedia.org/wiki/Brans%E2%80%93Dicke_theory]
So, there is something about General Relativity that makes it especially favoured. That is mainly that it has proven to be extremely useful and accurate. It has made predictions and they have held up well in reality. None of that will change if some experiment discovers a minor breech or failure of the EP.
We already know, or strongly believe that General Relativity (GR) has limits and is only an approximation. You are probably already aware that GR doesn't match up with Quantum theories well and you've probably also heard that it is unlikely there is a genuine singularity in a black hole. So, if something did happen, some experiment managed to falsify the EP, then yes... that does establish that GR can't be exactly right BUT so what? We knew that already. GR is just an approximation but a very good and useful one and it will almost certainly remain that way for many years.
Newtonian mechanics is only an approximation and a less accurate one than Relativity but it is still so useful that we continue to teach it in schools and universities.
Best Wishes.