Two way finite state Automaton NP Compleness definition clarification?
For a problem to be in NP, there needs to be a polynomial time verifier for it. This polynomial time verifier need not be the "obvious" one. Ruling out one specific verifier doesn't prove that the problem is not in NP.

For a problem to be NP-complete, it needs to satisfy two properties: (i) it needs to be in NP, and (ii) it need to be NP-hard. You can look these concepts up in many sources. In particular, the fact that there is some natural exponential time algorithm for a problem doesn't imply that it is NP-complete, though it is true that every problem in NP has an exponential time algorithm.

Unfortunately I am not versed well enough in two-way automata to answer your specific question, but my comments above imply that the answer doesn't necessarily have a direct bearing on whether the emptiness problem for 2NFAs is NP-complete or not (or even in NP or not).