Take their offer
"Every time I've gotten one, the PC began by asking for my driver's license."
This is the way that it has worked for me as well. What works fairly well is to initially put-off their offer by saying something like "I never play long enough to get any decent comps anyway". When they insist, say "OK", but make it clear that you don't really care much one way or the other.
After this set-up, the PC will invariably ask for ID. Offer something else such as a credit card, etc in another name (note that this is not "identification"). If they balk, then make up a plausible excuse for not having your ID on you, and offer to fill out the form with address, etc.
I have also found that hosts are alot more agreeable to not requiring valid ID to issue a VIP card, so you may want to make a large buy-in and then immediately request a host. This works even better if you can ask for a host by name and say that another player recommended the host (use a name that the host recognizes).
The key is to get the PC/Host more worried about pissing off a good customer than about failing to identify a card counter.