Thanks for the thoughts everybody. I ended up deciding to drive to Hatta through Oman. Didn't bother getting insurance for the rental car, though I knew it was a risk. I was prepared to turn around at the border crossing if necessary but they only checked my passport. Driving back into the UAE the soldier just asked if I was a tourist or a resident and waved me through. Didn't have to get out of my car for either checkpoint.

The way back to Dubai from Hatta I took the longer way around Oman. Both drives were incredibly easy (outside of Dubai.) The scenery was great, and it was fun to see camels along the side of the road in the desert.