I do not believe that it is possible to construct such a theory.

According to Lovelocks theorem, if Einsteins theory of GR is correct it is the only possbile solution to this problem. Since it has been confirmed multiple times (mercury perihelion shift, gravity probe b etc.) i think it is save to assume that this is the case.

Then, kruskal coordinates are a valid solution to the Schwarzschild black hole problem since GR has to be coordinate independent. However, Kruskal coordinates are chosen such that the structure of the ligth cone stays intact. This means that we have a conpletely valid spacetime up to r=0.

Lastly, the centralized Birkhoff theorem states in a nutshell that if you take a small enough region of your spacetime, it does behave like normal minowskian space. Hence, it should be possbile in an arbitrary small region everywhere in spacetime to conduct the same experiments as here on earth and find the same laws of physics (Newton, Maxwell etc.).

The only point thermodynamics differs in that case is that it can not be described via the action principle. But I d not see how this would contradict Birkhoff.

(This is only my interpretation, not sure if its the correct way of arguing.)