NP-completeness of graph isomorphism through edge contractions with an edge validity condition
The answer is yes.  Suppose that you had a polynomial-time algorithm to solve your revised problem.  Apply that algorithm to the special case where the subset of edges available at each step is the whole set.  We'd then have a polynomial time algorithm for the original, known NP-complete, problem.

To have any hope of dreaming up a non-NP-complete version, you'd need to put a restriction on the subsets that were eligible that would prevent an immediate reduction to the original problem.  At that point you'd have a new problem which you cannot easily show is NP-complete.  However it is in the nature of NP-complete problems that, if your restriction avoids 2+ choices at each step, it very likely to be NP-complete.

(Note, it is currently unknown whether the simpler problem of identifying whether two graphs are isomorphic is NP-complete.)