What is the flaw in this model for homomorphic encryption?
The last paragraph of Section 2.3 of Gentry's "easy" intro to homomorphic encryption, I believe, contains an important result which applies to your model.


  Researchers [1, 8] showed that if $\mathcal{E}$ is a deterministic fully homomorphic encryption scheme (or, more broadly, one for which it is easy to tell whether two ciphertexts encrypt the same thing), then $\mathcal{E}$ can be broken in sub-exponential time, and in only polynomial time (i.e., efficiently) on a quantum computer.


Where 1 is Algorithms for black-box fields and their application to cryptography and 8 is Quantum algorithms for some hidden shift problems.

Since your isomorphism is deterministic, this would most definitely apply. Granted, just because it can definitely be broken in sub-exponential time does not mean there is no hope (RSA can be broken in sub-exponential time), you are going to have to give a very good hardness proof to get anyone to use your system.