I was at Big Bend in the middle of February two years ago. It is really nice that time of year. It is high desert, so, it will get cold at night... low 40s. With the right camping gear, though, it is really just fine. I camped in the Chisos Mountains campsite. It is bowl shaped, and faces south. This camping area keeps the warmth from the rocks in a little better at night than the others.
I am not familiar with the vehicle you are thinking of taking, but there are a lot of jeep roads there! You can go backcountry driving/ riding and camp out there. The border patrol has a few officers designated to the park, so no real need to worry about intruders. Beware there are BIG CATS out there. You will want to bring some kind of bear proof food containers.
The mountain lion was seen each night I was at camp... three or four nights I was there. I didn't get to see him, but they do exist there. Don't let the weather stop you.
Piglet: Because Hogs Are For Old Farts!