We've done it many times in our cars and once in our tow vehicle with 33' TT and it wasn't too bad truth be known, although it is quite a climb and you have to traverse some very curvy roads and inclines. If Artesia is your destination looks like you don't have much choice, take it nice and easy and you should be good to go.
As mentioned above, if you are not time constrained and enjoy beautiful pine covered drives you could continue on 54 to Tularosa and take 70 through Ruidoso which has a number of wonderful campgrounds and a resort casino for golfing and dining. Take 380 to Roswell and down 285 to Artesia, both are easy drives when towing.