A private car is very unlikely for the kind of tour group you describe, though I suppose technically possible. There's basically a fleet of EgyptAir buses that pull up at ABS, take all the people who just arrived, ferry them to the site, pick up another group, take them to the airport, pick up the new arrivals ... and so on through the day.

As someone upthread said, I suspect your CAI-ABS flight stops in Aswan, where you can leave your luggage and pick it back up before Luxor, and that's where you're going to want to leave everything. I speak as someone who wound up carting a laptop through Abu Simbel.

If you have several hours in Aswan, my guess is that your group will probably be herded along for the mini-tour of Aswan that they do, including seeing the dam, and possibly a felucca ride, if you have the time. It's not bad.

I'm not surprised the last flight out of ABS is at 1pm. The heat is pretty unbearable even at 10am.