What is the flaw in this model for homomorphic encryption? Imagine a Field Isomorphism  $g : \mathbb F1 \to \mathbb F2$ given by some $g(x)$

Assume a client is planning to outsource his computations to server, translates every possible $x$ as $g(x)$ and sends to server and once he gets the result he translates g inverse to get the answer.

I know the above is straw man solution for Homomorphic encryption, but am being naive to think of possible problems with such model. One problem I could think of is simple frequency analysis would break the system (but this can be mitigated by coming up padding schemes that retain the homomorphic nature), but what are others?

The catch here is , Most of the current homomorphic encryptions are trying to encrypt the data and perform operations on the encrypted data, but here instead of encrypting it , just the input set is transformed to a another field and such mapping $g(x)$ is kept secret.
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.