As far as I know, no such FAA regulation exists. Like others here, I have also ridden in a hotel shuttle with airline crew.

However, I wouldn't be surprised if the airline crew commandeered the shuttle for their own use (not literally, but demanding that they be served separately).

One time I waited for a scheduled hotel shuttle early in the morning, and I found out that some airline crew showed up 15 minutes early and demanded to leave NOW. Why, they just couldn't be bothered with the posted schedule (the one that says "5:00 AM") that the rest of us live by.

I was very mad that the hotel let them do this, and I swore I would never stay there again, and I've kept my promise. I suggest you do the same.