[Bakkot]

Why not use the potentially exponential algorithm initially and timeout and fall back to the linear case if the sometimes-quick one has taken longer than the linear one would have? At most a 2x slowdown for complex cases, and substantial speedup for the simple ones.

[microtonal]

It depends, if you have the opportunity to build a DFA once, why not do it? It's only a one-time cost.

I think the more serious problem is that what most people consider to be a 'regular expression' does not express a regular language, and hence cannot be expressed as a finite state automaton.