Going to Big Bend NP is probably my #1 destination with the Minnie Winnie. I've been to most of the camping options in and around the park. Of course my favorite choices are the Chisos Basin CG, and primitive camping (Grapevine, Paint Gap, etc.) along Panther Junction Road. I specifically target Feb-April and Oct-Dec for those places. However, as noted in the other posts it's quite hot in the August/September timeframe. You'll certainly want to have electrical hookups to run the A/C, and your choices are Rio Grande Village RV CG, and several campgrounds in Terlingua. Since you have tow vehicle, my choice would be Terlingua. There's several decent places to eat, and the area is a good launch point for day trips into the park. The Rio Grande Village CG is okay - just a big parking lot with those 90 degree L-shape sewer connections with water/30a/50a. However it is very convenient to key spots along the Rio Grande.
As noted in other posts, there's Boquillas Canyon and the crossing to Mexico. If you plan to cross over into Mexico (which is highly recommended), you'll definitely need a passport or the "passport card". They won't let you use your driver's license anymore. The last time I went there, I forgot to bring my passport, and the border officer explained the international policies when I asked to use my Texas driver's license. Even if you don't have your passport, there's a lot of other things to do. Be sure to check out the hot springs along the river, and head up to Chisos Basin. Another one of my favorite places is along Ross Maxwell Scenic Drive. Along that route is the Sam Nail Ranch, Mule Ears Viewpoint, and Tuff Canyon Overlook. Just talking about it, makes me want to start making plans for my next visit.
I've also stayed at the Chisos Mountain Lodge, with my car, and had no complaints. The on-site restaurant is decent, but keep in mind the logistics involved with transporting food. So maybe not gourmet, but they usually have something good to eat. Also there's a number of super nice trails around the Basin area. You may want to head up to Pinnacles, or if you up to the challenge - go to the top of Emory Peak.
Enjoy!