Hello,
Due to work, I have been commuting between El Paso and the Carlsbad area every weekend for the past few weeks. This route is fine and I see many motor homes and trailers going both ways. There are climbing lanes on most of the grades to make things easier for larger vehicles.
The climb up near Guadalupe Mountains NP is long but just back down and take your time - the semi's do it all the time and I have seen some significant oversize loads doing it. Just be aware that there can be some strong crosswinds along this climb if the weather is changing. Again, just be aware.
If you have time, I would strongly suggest stopping at the Guadalupe Mtns visitor center to have a look. There is good easy access from the highway and a large parking lot with room for your MH.
No personal experience with the RV parks in the Carlsbad area but most say that the KOA on the North side of the town of Carlsbad is the best choice. There is a park at Whites City which is just outside the caverns NP but many say it is not too good.
It is interesting to watch the scenery change as you climb up from El Paso and then crossing the salt flats below the Guadalupes. Enjoy the drive and have fun.
Let me know if you have any specific questions.
Terry