You definitely do not need a gun at Big Bend. That's silly.
The hairpin going into Chisos is tight, but doable. The hiking there is the best, but the camping is better out in the park. If you're at all adventurous with your trailer, there are plenty of backcountry sites that you can get to with a 2WD and your Oliver. You might check with the backcountry office when you arrive to see what's available and if you want, scout them out with just your truck before committing. We usually go right after Christmas, when there are people camped outside the office to get spots, but when you're there you should have a good choice. Generally the roads are bumpy but if you stay off the designated 4x4 trails, then no problem - just some long drives to get to some of them.
Last time there, we camped at the gravel pit - not great scenery, but you're right on the Rio Grande. Our favorite spots are along Pine Canyon and Glen Springs road. A handful of spots there are far enough in to feel secluded yet don't take half an hour plus to get back to the main road.
Pay attention to your gas while you drive around. It's a big park so don't let yourself get too low. There's a good roadside BBQ spot in Terlingua if you get tired of camp food. Boquillas entry is open again, if you want to take your passport. There's really not much to do there but eat some average TexMex, but it's a poor town and they appreciate the patronage.
If you've got a telescope, it's worth packing it - the night sky is pitch black in the backcountry.