Hello Dave ; I've been driving the route south through MT, Utah and Nevada for 10 years. Seems like every year no matter how early I leave , I find snow somewhere. I usually try to arrive south by Nov. first.
Vegas is a safe bet for no snow 99% of the time in Nov. ; but I've got into a
snow storm in Searchlight NV. in Dec. But considering it was -23f that day when I left Ontario Oregon , a skiff of snow was the least of my worries.
Put on 750 miles, that day trying to stay ahead of the storm.
I consider my coach a 28,000 lb snow magnet.
No matter what the Vegas and Phoenix , chamber of commerce say , there can be cold in the desert south west , we usually get a 3/4 day stretch of overnight frost in Jan. around Lake Havasu City , and you should hear the whining.
Once in a while there are even reports of frost in Yuma.
BTW; I prefer the route south through Nevada to avoid SLC and the 80 mph speed limits on the freeways.
EDIT: If you only have 6 or 7 weeks to spend south , I'd try to get last week of Jan , Feb. and first week of march .