Why wouldn’t you bill directly? Why wouldn’t you travel with your own recovery nurse to help recover? Why wouldn’t you require a proper medical screening? Why wouldn’t your make sure the proper equipment is there before doing a case? If you’re an anesthesiologist traveling to a dental office, it is on you to make sure all of these things are up to standard.
I suppose one could do all those things, if one could find a dentist agreeable to the arrangement. That's ... uncommon.
I have a very negative opinion of "dental anesthesia" gigs in general because every single one of them that I've been exposed to has been some fly-by-night clown running a fabulously unsafe clinic, charging patients high fees for anesthesia, and looking for some desperate anesthesiologist to come work for them.
And every few months we have yet another thread here about yet another dental sedation death that fits that scenario exactly.
I get my own dental care done at a very nice clinic. They have an anesthesiologist come in regularly (weekly?) to do sedations. It's some independent guy ... older guy, near end of his career. (My group doesn't do that work and isn't interested.) I don't know what his arrangement there is, but I know they don't have a PACU and I know he doesn't bring anyone with him. I'm a little dubious about the whole thing, based on the little I've observed and been told.
The responsible dentists I've known take their patients to hospitals or ordinary ASCs for dental restorations under anesthesia. We do a bunch of them here in our hospital-attached surgicenter. A lot of them are special needs kids. That is fine.
As you well know, there are very, very few people like you (dentist anesthesiologists) out there. You'll be qualified, well trained, and safe once you finish training. But you're not the norm. I don't know what % of dental sedations or general anesthetics in the USA are provided by dentist anesthesiologists (maybe you have numbers) but it isn't most of them.
Google tells me "The American Society of Dentist Anesthesiologists (ASDA) has approximately 500 members in the U.S. and Canada." That's not a lot.
The point of my post above is that dental sedation/anesthesia in the USA, when done outside of hospitals/ASCs, is often a very shady, very dangerous business. You're not part of that problem, and good for you. I'd still caution any anesthesiologist to be very, very wary of any dentist trying to recruit them to do in-office cases.