You could register it in your home state as the link shows.
Or could get a drivers license in NM maybe. When my license expired while working away from home in AZ, I went to the dmv, used the address of the extended stay I was living in and they gave me a license. (Still have it. Good til like 2037!). I brought down one of our personal vehicles and registered it as well, before I really had a residence there.
I've also registered vehicles in several states with a drivers license from a different state. Might have to do a little talking to get that done though.
