Christmas can get busy at Big Bend so my recommendation would be to make reservations at RGV (the NPS campground not the rv facility). Site 23 is a good one. On that end of the Park, you can hike the RGV nature trail and the Hot Springs from the campground then drive to the parking area for Boquillas Canyon (say 'Hi' to Victor the singing Mexican for us).
Parking lots off the paved road provide trail heads for several hikes mostly on the west side of the Park - Burro Mesa, Tuff Canyon, Sam Nail ranch, Santa Elena Canyon (don't miss this one).
We tow a 19' 5th wheel into the Basin with no problem and I've seen tour buses at the lodge so vehicles larger than yours do make it in. If you get there, the Lost Mine trail and the Window are both strenuous but worth it. If you're really up for awesome, try the South Rim (14 mi RT). Even if you don't make the whole route, its a great experience. Site 47 slopes a bit but is our favorite there. We've been able to transfer from one campground to another without any penalty in the past if we found a site we liked better.
We'll be there in early October for ten days.