Personally, I would not feel comfortable nor my hubs with towing into the mountains.during the winter months...and yes you will be climbing elevation to get there! We've towed thru a foot of snow years back on a trip from Michigan to Texas and it was not fun! Our truck is 4x4, handled the snow beautifully even in Kentucky....but I believe you may have a bit bigger climb in & out of Ruiodosa. We camped in Ruidoso 2 years ago. It's beautiful there!! However, you need to consider what conditions you might unexpectedly be dealing with and if your TV will be able to handle it.
In the best conditions, go for it.....but should that change be prepared! Going from Texas, again depending on the weather and time of year you will need to watch the weather for any fronts that come down.
I grew up in Houston, made many trips to Colorado going thru New Mexico for winter skiing and the roads can get dicey fast!
Whatever your decision, be safe have a great trip and enjoy your vacation!!!