I noticed you said BB area and not just BB NP.
RE the NP: I agree with others that you'd be best to camp in the regular no-hook-up CG at Rio Grande Village. You can run a generator as needed during the day, to make coffee, microwave, whatever. Be advised there is a no generator section. If you need AC then you'd be much better off at the parking lot type full hook up CG by the store. You'll be driving the C to the trailheads and I wouldn't be surprised if it amounted to an hour or more per day of driving, which would help keep your coach batteries charged. That's a good thing since most RVs don't charge quickly enough via their genset running their built in converter/charger. But that's another story.
You can cross into Mexico at the primitive crossing just established near RG Village. You might be able to find someone to take you trail riding (horses) over there, but that's just a guess. They certainly use horses a lot on the other side.
At RG Village, you are at the far end of the park. I think with a 7 year old, you'll not be biking too many places. The trails are not suited for it. You can bike over to the store for ice cream and internet, though. :)
As mentioned, the road to the Chisos Basin is considered to be too tight for anything much over 25'. It's really the last mile or two dropping down into the basin where the problem lies, IIRC. So even if you can't go the whole way down there, you can visit trail heads and such along the way.
There is a BB State Park to the west of Terlingua. There are no CGs in it, but it's good scenery and hiking. It'd be too long of a long day to visit there out of a CG at RG Village in the NP. No way. Instead, you might enjoy spending a few days in Terlingua. At first it looks like not much. But it's quite the place. I'd recommend BJ's RV Park. Same commment applies. :) They are very nice and super helpful there. They will tell you where to go, which nights are better at which eatery, etc. Also, from Terlingua, you can go back into that side of the NP to do the trails etc. There are some back roads you could bike on from BJ's. They just go off into the semi-civilized scrub outside of town, but it'd seem exotic to your kid.
Davis Mountains SP is pretty nice, but I read here that its CG is closed for repairs this winter. You could check to make sure, because last winter it was their lodge which was closed for repairs, not the CG. There is a private CG in Ft. Davis as well. It's a nice little western town. Nearby is the observatory, with a very nice star gazing show. Bring jackets.
Marfa is supposed to be a neat town, but we didn't get there.