I've known a couple of people who have gone through respecialization programs and gotten licensed just fine in multiple states. My advice would be to look at the licensing laws in the specific states that you are wondering about though and possibly call the state boards if the language in the laws is not clear. Some states specifically list APA accredited programs or those deemed "equivalent" as making you eligible, some may specifically list respecialization programs as fulfilling the criteria, and then others may not. The place where it likely gets more difficult is if the licensing laws say that you need "a doctoral degree from an APA accredited program" (a growing number of states are stating this) without any kind of qualifiers about other programs or respecialization. In this case, it would likely be difficult to get licensed without the degree itself coming from an APA program, but again, call the specific state boards that you are wondering about because they are all different.
In terms of advocacy, it really comes down to you. DCTs are generally responsible for telling you whether a program (including any formally recognized respecialization programs that they supervise) will qualify you for licensure in that state that the program is in. But it's likely they may not be familiar with requirements in other states.
Sorry for the lack of specifics. Maybe others on here will chime in, but it's my understanding that it's pretty rare for people to respecialize.
