https://pubs.nmsu.edu/_circulars/CR...ll fescue can be grown throughout New Mexico,
The climate in the Southwest is quite different from that in Ohio and South Carolina. It is quite a bit drier. Tall fescue doesn't do well in the south half of South Carolina. I found the article above from New Mexico extension that discusses tall fescue (appears to be KY-31 and not a novel endophyte) which says it does well in the north half of NM, especially with irrigation. Based on what is said there, I get the impression that the tall fescue will do ok in southern NM. The extension article also discusses orchard grass and perennial ryegrass in New Mexico. Depending on your location within NM, I think you mix will do OK to well.
As for the crabgrass varieties/species, I found a good bit about Red River. I think it is a good choice to overseed your cool season mix. HOWEVER, I would not overseed your cool season mix the first year and the second year would depend on the performance of the cool season grass up to that point. The cool season grass needs to get itself well established without competition. I'm particularly concerned about the orchard grass from this aspect. Also, the ryegrass is short lived (for a perennial) and I would want to maximize the production from the ryegrass before it plays out. I think that overseeding with the crabgrass would introduce some competition for moisture (minimal though) with the ryegrass which might hasten the ryegrass playing out by a year or 2. I cannot find as much specific information on the mojo crabgrass as the red river, but it appears to be a good producer and I would expect good yields from it.
I would check with NM extension further for more information.