If the national organization really does have a strong and well-developed identity program, that program will also address the issue of how to incorporate that identity at the local level.
If their national identity guidelines don’t do this, much might depend on the organization’s culture and how closely other local chapters tend to adhere to those national guidelines.
If it were me, I’d bring up the question with the local chapter and then follow their lead on how tight they think those parameters are.
It could be something as simple as adding the words “Timbuktu Chapter” as a tagline to the logo or making sure that any text or headlines help make the distinction clear in those instances when it needs to be clear. On the other hand, they might tell you to pay lip service to those national guidelines, but not to slavishly follow them. It really should be their decision or it could come back to bite you when the subject finally does come up, which it will.
Bringing up issues like this with clients this is part of your role as a designer and consultant. Remember, design is probably more about problem solving and considering client objectives than it is about aesthetics.