Is unification over regular expression equations doable?
I'm fairly sure this is possible. This seems to me as a special case of set constraints over tree languages: we can view regular expressions as a restriction of regular tree languages where each node has 0 or 1 children. These can handle union, concatenation, and recursion (star), and you can solve for variables like you describe.

They're decidable, even when we allow negation. You can see a great survey here:  https://theory.stanford.edu/~aiken/publications/papers/ppcp94.pdf In general, Alexander Aiken has a lot of great papers on set constraints.

You might find more of what you're looking for with "word constraints" or "set constraints over words". The complexity situation isn't great for trees, but it might be better for words.