By my reckoning, Burning Man takes place about 50-60 miles south of Prime Base*, "right outside" only on a national scale. And I see no reason why anyone at that location would choose to go further into a barren desert towards Prime Base, north is literally the hardest, least promising direction they could go. To be honest "refugees at the gates" was always a stretch when you consider the obstacles to travel in the region and that roads past it were hardly on the way to anywhere in particular.
Remember also that Prime Base didn't even unbutton until a couple of years post-war*, and what are the odds that Burning Man staffers would still be in the Black Rock Desert after all that time? These aren't really rugged survivalists, why didn't they head east or west towards towns and modest civilization? Prime Base could unbutton right after the war, notice a small group of people hours away, and risk the entire Project in an environment that could not possible have been adequately surveyed yet... but why would they?
*: Per 3ed, at least, since I do not have 4ed.
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