Might want to search for native raspberries of NM.

My guess is that any type would do well under the conditions you've created... I've seen quite a few healthy raspberry patches in some hot places, but mine are along the north side of the house in an area that is not as hot as Albuquerque. It would be very interesting to look into NM native raspberries as Flattie said, the red raspberry grows everywhere in the Sangre De Cristo mountains, and I've read that blackcaps (black raspberry) are native to NM as well. Blackberries also do incredibly well here with enough water.

Oh yeah, there's that raspberry farm down there in Corrales, maybe you could give them a call. Good luck!