I have driven those roads and 70 is fine to Alamogordo. The main issue I see is taking 82 out of Alamogordo which leads you through Cloudcroft and the Lincoln National Forest.
It is quite a climb with many curves from Alamogordo to Cloudcroft and for someone who is towing it is not an ideal route. Once through the forest the rest of 82 is OK and 285 from Artesia to Carlsbad is not a problem.
Not sure where you are staying if visiting the Caverns but an alternate route would be to take 10 to El Paso (time it to avoid rush hour) and take 62 to come up the backway to Carlsbad. I'm not saying 82 isn't doable, just tasking for both driver and truck.
