I'd expect they used a stolen card (or a stolen card number, more likely). There's almost no way anyone could be stupid enough to do that with their own card.
The interesting thing will be the gmail account's IP history. I suspect SFPD will get the IP history and I hope this DJ Pattrson logged in from an identifiable location (or that the phone number, presumably a prepaid, is linkable to location and identity somehow).
SFPD is one of the better police forces to have to work with on this; I'd sure rather be dealing with them than Oakland or various other East Bay police forces. You might also want to involve REACT (http://www.reacttf.org/), the Silicon Valley (incl SF) high tech crime task force. They'll be good at tracking someone down through online identity, compared to local police.
I wonder if AirBnB has offered the host/victim some AirBnB credit to stay elsewhere while her place is being de-trashed.
I check into lots of hotels and about 50% of the time they require an actual government-issued photo ID; I assume for room-trashing situations and the probably more-common situation where somebody racks up a huge bill and then steals away in the middle of the night leaving nothing but a stolen CC number.
It's a lot more difficult, and the legal penalties are much steeper, for forging a government-issued identity document than for using a prepaid credit card with a false name on it.