lol
I did Utah in October – it was damm hot
If you get to Arches/Moab then you will be able to hire a bike, In June Slickrock will be unbearable, I was out there about 8-9am and it was stinking hot. You can do the Enchilada as a day shuttle and is one of/if not the best days on I bike I have ever done. So if you are going to have a day pick that one 😉
For a few days riding not sure it’s worth taking one though, unless your in a breaking bad size RV you will be getting the bike in and out every time you want to have dinner etc.
Not sure of your routes but Park City area has some great lift assisted riding, is way more chilled than most of Utah and has all the old winter Olympic stuff, think you can ride the bob sleigh track and stuff like that.