I've done the UTBDR by motorcycle. I don't think it would present much difficulty for a GX with proper tires and running boards removed, but know that conditions are highly variable. Perfectly smooth roads can become treacherously rutted out with a single storm, so having nav capability for bailouts is advised. Lurk on some of the moto-oriented sites to gather timely beta on conditions.
If you haven't been to Monument Valley/Valley of the Gods I'd recommend you budget an extra day for Section 1. There's so much to see just off the route. Butler Wash and the Abajos (both on the route) are magic from afternoon to sunset--in the summer the clouds build in the afternoon offering a reprieve from the heat and stunning vistas. There are plenty of other magic spots as you continue north, but I would recommend you end at Bald Mountain Pass or Evanston. That last 100 miles is fine, but nothing remarkable. You could also skip the section between Green River and Wellington if you wanted to make up some time.
With a normal snowpack, the earliest you could expect to start is late June. We're above normal this year, so I would say end of June through early October is your window. If you do go in the summer be careful of and prepared for extreme heat. People frequently die when they get lost or stranded on otherwise simple outings. The desert DGAF.
COBDR is my favorite and doable in a GX.