Regular expressions without union but with backreferences
If you are allowed to use complementation to express $\Sigma^*$ as $\emptyset^{C}$, the resulting language class includes the pattern languages (see, e.g., this survey by Kai Salomaa).

As shown by Angluin, the membership problem for pattern languages is NP-complete. (This holds even if you consider pattern languages that are generated by patterns that do not contain terminal symbols.)

So, membership for this class is as hard as for the full class of regular expressions with backreferences.