My spare is factory-mounted on a rack attached the square bumper hanging off the rear of my small rig. The bumper also holds the stinky slinky.
But the "should you" part of the question really focuses on whether YOUR bumper is designed to handle that load.
For comparison, my bumper attaches directly to the frame rails. The bumper itself is approximately 4" square tubing with about 1/16" wall thickness (or more). The attachment points on either end are welded to the bumper, and then the attachment is bolted in two places to the frame member (on each end)...with, I believe, 1/2" "Huck Bolts." It's designed at the factory to carry not just the weight, but also the twisting loads imposed by the spare tire and carrier standing well above the bumper.
In this way, my spare tire rack is much like a bicycle rack mounted to the rear bumper of an RV. Ever see one of those things with bicycles being tortured to death as the RV bumps up and down? It's a hard life being a bicycle back there. But the spare tire mount is far stronger and the system is much more stable...if the bumper/frame mount is up to the task.
What could be a problem? A "C-channel" bumper is not nearly as strong, no matter how it's mounted. And it won't withstand the torsional twisting imposed by that much weight and leverage from the spare mounted high above the bumper. So it depends on what's on your rig.
I got a peek at what I think your rig is and your bumper appears to be up to the job, assuming it's like the one in this video at about 14 minutes in:
That one even seems to have a Class II receiver under the bumper...mounted to the frame. So your prospects are good. In this case, I'd move the spare mount as far as reasonable to the left or right side of the bumper to stay out of the way of the hitch.
One other consideration is weight and balance. Assuming the tire/wheel/mount combination come to about 125 pounds or more, you may have the small consideration of whether moving this weight from the front to the back will lighten the pin weight too much. Unlikely, but worth considering. A rig that has a factory trailer hitch suggests that this won't be a problem.
Just be sure to buy a decent spare tire mount. My factory mount looks a lot like this. https://www.etrailer.com/Spare-Tire-...lson/TR-1.html
As for sun damage on the tire...buy a cover. https://www.etrailer.com/s.aspx?qry=...are+tire+cover
I see bare spares all the time, but there's no excuse to leave the poor tire exposed to UV when you can protect it for $30.