OK, it turned out to be a false alarm. I went back to Avis today, with passport as a second form of ID, but this time they accepted my license as ID. Turns out yesterday the agent either typed in or didn't type in my middle initial, not sure which, but it confused their system and that's why my license didn't get accepted the first time.
I do have the star on my license, BTW. Someone actually told me about that yesterday.
Thanks again for all your concerns!