I think your nuts to think you can do "good mtn biking" in Big Bend in August.
You would be best leaving the bikes at home and bringing hiking/backpacking gear instead. At least that way you could be up in the Mtn's where it's relatively cooler. Please understand the Park in August commonly gets over 100 degree's and bikes are only allowed on the dirt roads (not on trails) and the dirt roads are very rocky and rough. Throw in a few flats from cactus spines and you have a bad experience for everyone. Big Bend is a great park but not a great place to Mtn Bike. If you want to get off paved roads you have a few options: River Road East or West, Old Ore Road, Paint Gap Road, Maverick Road, and a few shorter routes but none of them are loop rides and all will be very hot (dangerously so) and probably sparsely travelled. They have already found 3 bodies in the desert this year... TWWG
Of course, the Big Bend Ranch State Park has more mountain biking opportunities than the National Park, but I'd still concur that August might not be the best time of year for that activity. I'd suggest contacting the staff at the state park and asking for their recommendations.
