Is this DH variant a workable zero knowledge proof?
why it might be problematic to do this in practice.


One issue with it is that it potentially allows the attacker an Oracle to the cDH problem.

For example, suppose the Prover uses his $g^x$ value as his key share with a DH key exchange with Alice, who sends $g^a$, and both sides arrive at the $g^{ax}$ secret value.

Then, what your protocol would allow an evil Verifier to do is ask the Prover to prove that he knows $x$, and submits $g^a$ (or a blinded version) as his challenge; the Prover sends back $(g^a)^x$, and that gives away the secret value.