Well, you have a 2 hour time zone working to your advantage to get your daughter up.....
Regarding the tires, that’s why I mentioned AWDrentals or Turo (or other private car rental options), since indeed, I don’t know of others that offer snow tires.
The good news is with Rugged rentals if you have a Jeep or such, you hopefully get all-terrain tires, and some of those are 3peak+snowflake rated. This ups your odds of being allowed up the canyons.
However, this is still no guarantee:
“Enhanced public safety restrictions when yellow lights are flashing
During extreme weather, and when UDOT and Law Enforcement determine it is in the best interest of public safety, traffic may be restricted to mounted tire chains, studded snow tires, or Four Wheel Drive vehicles. Four wheel drive vehicles are not relieved of the snow tire requirement although most all terrain tires are acceptable.
Unfortunately, a small number of tire manufactures label their products as M+S (mud and snow) when in fact they do not possess the aggressive separated tread needed for safe canyon travel. This is especially true in some “sport –performance” wheels seen on luxury SUV’s.”
http://updsl.org/page_canyonAlerts.php
(Note that although they write M+S, I think they actually mean 3peak+snowflake.)