Much of UT-12 is Winding 2-Lane Road (some of it with at least a narrow shoulder - but some without). With any length of Trailer, you'll likely make an average of 30MPH - not counting all the times that you'll WANT TO Pull-Off in one of the 1/2-dozen Scenic Overlooks or HAVE TO Pull-over onto whatever serves as a shoulder to let the accumulated Faster Traffic get past. If you are just Thru-Driving, one would probably suggest that you "Go Around".
But, if you do, you'll miss some Very BEAUTIFUL Country!! This is Red Rocks Country at its Finest, with some Alpine Driving bundled in...
I'd really suggest that you Don't want to Miss It... But don't plan to Tow through it without setting aside AT LEAST a Full Day!!
Better would be to Drop the Trailer at an RV Park near either end.
At the North End, you could try Torrey Wonderland RV Park (never Camped There but used the Dump-Out because I'd been Boondocking in the area) or Capitol Reef NP CG.
If you set aside a whole day, catch Capitol Reef at Sunrise. Then Drive the Loop south along UT-12 to Boulder. From there, take the Burr Trail East through Long Canyon into the southern tip of Capitol Reef NP, possibly stopping for a quick Overlook Hike at Lower Muley Twist Canyon to the Strike Valley Overlook (try to get there before mid-afternoon), before taking the Burr Trail Switchbacks down to the Notom-Bullfrog Road, and that back North to your Camper at Capitol Reef.
At the South End of UT-12 is Bryce Canyon and Red Canyon - both of which have Great Camping. And of course one could spend a day or two at Bryce and still want MORE...
The problem with that Plan is you miss the middle part - Boulder-to-Escalante - driving along the HogsBack and other Slickrock areas. You miss the easy Dayhike of Lower Calf Canyon, or the somewhat more strenuous one to Escalante Natural Bridge.
Any of the Above, the Suburban will do with Confidence and Comfort... Not so much so with the Travel Trailer...
Pup: '11 HW-296
TV: '09 RAM 1500 Hemi 4x4 CC