There are a couple dozen FHU 50amp sites for use in a parking lot next to the Rio Grande Village store. These are concessionaire run sites. Check the Big Bend web site under camping for info.
The other FHU options outside the park all involve at least an hour drive to parts of the park you will want to visit.
The motor carriers atlas doesn't show the roads as 'good' because they are considered toll roads. i.e. you have to pay an entrance fee to get into the park - so that makes it a toll road. As mentioned above - the main paved roads within the park are excellent.
Be careful with unpaved roads in your toad. Check with the Visitor Center about specific routes suitability for that day. Some back roads in Big Bend could be too rough for your Jeep.
I would recommend you take US-90 west from San Antonio vice I-10. A much more interesting drive. Turn south into the park at Marathon.
Leaving the park - TX-118 to Alpine is an excellent road.
A much more interesting road is TX-170 to Presidio and then north to Marfa. There is one very steep section, but I've driven it several times, once pulling my 36' TT.