Your best bet is probably up in the Sacramento Mountains and Lincoln National Forest east of Alamogordo. Lots of public land up there. There is a campground (Aguirre Springs) up in the Organ Mountains between WSNP and Las Cruces, too, but not sure about dispersed. WSNP is pretty much surrounded by the missile range (WSMR) and Holloman AFB, both of which are limited access. There is some BLM land nearby the park, but it will be pretty barren.