First, DO NOT BUY CHAINS! If the Interstate you plan to travel has chain requirements in place DO NOT DRIVE! Tomorrow or a day or 2 later the road will get better and you will be able to travel safely. Leave earlier than you need and allow plenty of travel time.
The short route is 15 to 70 but you have an 11,000 foot pass (no problems if the road is dry) and you may have to wait for it to clear. What I would do to ensure the best chances of a safe and easy drive would be 10E to Las Cruces and then 25N all the way in. It will add about 200 miles to the drive but at least half the trip will be in areas that never get snow or get a bit snow which quickly clears. There are no huge passes to deal with and while the I-10 trip has some pretty boring scenery, the I-25N drive is beautiful.
I have done both in the Winter and both will work if you watch the weather forecasts and either go or stay based upon that.
You will also be able to stay absolutely toasty warm in your RV and the tanks will not freeze if you camp in cold weather if you follow the procedures your coach was setup for. That mean you are going to burn A LOT of propane since the house you are visiting will not have the necessary 50amps to run some electric heaters. The furnace will blow hot air into your wet bays and keep everything warm there as well. If you would like some added piece of mind you can put 1 or 2 150W light bulbs (if you can find them) in the bays as well. It's pretty well insulated so a little heat goes a long way. It it's below 32 outside when you are driving you will need to run the furnace at that time as well. I spent last Winter in Detroit with temps sometimes dropping to -25. I had 50amp so I had electric heat in the coach, lights in the bays and I spent about $700 on fuel! Had a great time! I also bought a little weather station with a remote temperature sensor which I placed in the wet bays to ensure I could see if everything was staying warm enough down there to prevent any issues. Make the trip, have a great time but use some common sense and drive when the roads are clear of snow.