I rented a car in Austin this February and drove to Big Bend. I had about a week as well. The drive is about 8 hours in total, but there are some convenient stops on the way, like Junction and Fort Stockton. Also, you can drive through the Hill Country and stop in Johnson City and Fredericksburg (I stopped at LBJ's home in Johnson City as I'd just finished the four Caro books on him and was part of the reason why I planned the trip like I did - highly recommended to anyone, but not as highly recommended as Caro's biography of Robert Moses!).
In my weeklong trip I was able to do an extended solo Outer Mountain Loop hike, a separate hike to the mule ears, smokey creak junction and lots of other small hikes in the park. I didn't have a high clearance vehicle, so I missed out on Pine Canyon, which is one I hope to do in the future. It was all amazing!!! When not on the trail, I car camped at Paint Gap - which I really enjoyed.
Not 100% sure it will work, but here's a link to some of my pictures from the trip:
Anyways, Guadalupe sounds great too. I hope to get out there next, but I agree that El Paso is a much better starting point for that trip. I also wanted to make the trip work with Amtrak at first, but it just seemed like too many hoops.
One side note, I went to an REI in Austin before heading out and I would suggest having a good set of printed directions or a GPS. Around Austin, stores are often set back from the road in a way that, for me, made them difficult to see. The signage is relatively small. I can see why it's a nice thing, but it makes it hard for the first time driver there.
Also: you'll see if you drive through the Hill Country, that most of the land is fenced-off rangeland - beautiful, reminded me of Central Spain, but not great for secret camping... There are some nice state parks in the area though, on the llanes and pedernales rivers.