I visited the BB State Park in mid October, staying at some fantastic locations (Guale II is the best). The Rio Grande was/is flooded so driving from the State Park to the National Park was not possible without driving back to Marathon and then back down. As a result of the flooding, we had to enter through Presidio, not a popular tourist spot. The Park was virtually deserted! We took a 4wd and made a base camp, then hiked out from there. BB State Park has trails, but you are free to make your own, which we did.
Anyway, I have made Thanksgiving trips to the National Park, and the answer to your question is it can do anything. One year I had to turn back as an ice storm hit the highway. Another year, the weather was just great. If you are doing the Chisos, the the elevation will get you also. I would plan on lows in the 20's, heavy wind possibilities (we were almost blown off our spot in October). Prepare for anything, and take a compass or GPS, as it is really easy to get lost out there. We did some minor climbing down into some canyons, which were awesome. I love hiking the SW, and have done so from Texas to San Diego. Do call the National Park, also call the State Park as the Rangers there have more time to talk to you. They also are much more relaxed about what you do. No one cared about our bushwacking. They have several programs for cross country hiking at the National Park, the State Park is lasseiz faire about it.
Watch out for big cats, as we followed, or tried to, tracks of one we found in one of the canyons. I don't know what we would do with it if we found it, but it was fun while it lasted. The picture on my profile was taken out there.
Stay in Alpine before going down there if you need to stop. One of the great hidden spots in Texas.