Unfortunately, most hotels these days have a 24- or 48-hour cancellation policy, so if you book in advance and misconnect in London, you'll probably be out of luck. And any hotel (or airplane) can have bedbugs.

If you're willing to wait to see if you make your connection, I would probably try a Priceline "Express Deal," and specify only three- and four-star hotels with an airport shuttle.

If you have Hyatt points (or Chase UR points which can be converted to Hyatt points), there are two Hyatt Place properties near DFW, each of which offers a free airport shuttle and a free breakfast buffet; for this Saturday night, rooms are available for 8,000 points. (Hyatt has full-service properties at the airport, but they will be more expensive than the HP properties.)

If you misconnect in London, BA will be responsible for re-routing you, and for providing duty-of-care food and lodging during your overnight in London. (You could probably ask for BA's LGW-TPA nonstop if you don't mind getting over to Gatwick.)