I am not sure what your planned route is but if you are coming I-10 East from Yuma and then branching off at El Paso to take Hwy 62/180 then there is a significant grade thru Guadalupe Pass which is a few miles Southeast of Carlsbad Caverns NP. I would not expect any issues there but it is the only really significant terrain change along the way. Be aware that there can be significant crosswinds there as well.
If you do come this way there are several other natural areas of interest that you might consider as well as the caverns. The first you will come to is the Salt Flat area just West of the mountains. Not much to see really, but there is a roadside historical marker there. Then the Guadalupe Mountains NP which has the highest peak in TX. Both the Frijole Ranch House area and McKittrick Canyon are nice to see as well and are right off of the highway. Just about 5 or so miles Southeast of Whites City and the caverns turnoff is what I believe is a hidden gem of a picnic area. I would not try to take the trailer there but if you are without it, then Rattlesnake Springs is a really nice place. The spring there is the source of the water supply for the caverns facilities with the infrastructure being built as a CCC project back in the 40's I think. Anyway, there is a nice grassy tree shaded picnic area there that is a very nice diversion from the surrounding desert scrub landscape. Again, there is not really room for a trailer so you would need to be without it for this portion. Enjoy your trip.