This answer uses the same idea as the one given in a comment of mine on James Hanson's answer to [this MO question][1].

The basic idea is that one quick way to rule out the bi-interpretability between two theories is to show that some group arises as an automorphism group of a model of one of the theories, but the other since it is well-known that bi-interpretable models have the same automorphism group.

With the above in mind, it suffices to note that by removing Extensionality from $\mathsf{ZF^*+ GC}$ (equivalently $\mathsf{ZF+GC}$) we can easily build a a model of the resulting theory that has an automorphism of order 2 by swapping two urelements. But it is well-known that no model of ZFC can have an automorphism of finite order (basically because in the presence of the axiom of choice, let alone global choice, any set can "coded" by a subset of ordinals)


  [1]: https://mathoverflow.net/questions/456649/lists-as-a-foundation-of-mathematics/456652#comment1182864_456652