Right, so I went to the Chaco Canyon website and a concealed carry handgun is legal there (with the requisite CFL).
New Mexico is a bit puzzling because the usual rule is that ccw is legal in National Parks, Forests and Monuments in those states which issue such licenses.
But typically, ccw is legal in those same state's parks and forests. Apparently, in NM, a state that issues ccw licenses, carry is legal in the Nationally controlled park areas even though it is not legal in the state controlled park areas. I believe this is peculiar to NM (although there are probably some other states with a similar situation).
I have not been able to find any specifics regarding discrimination between resident and non-resident CFLs. Although NM and Nebraska both specify that Florida's CWL/CFL is only valid for handguns. (Florida's license covers "weapons" other than handguns, knives, etc,; that's the CWL part).