There is a very long and not entirely interesting story about how the Manhattan Project security force was assembled in Vincent Jones, Manhattan: The Army and the Atomic Bomb, chapter 11.

The short version is that the higher-up commanders and officers were more personally selected, but the day-to-day guards and security forces (and there were numerous, separate security forces) were just recruited from the same pool that you'd get security forces for any other kind of base.

I don't think you'd have been able to request a special assignment there. The low-level security people, as an aside, were given very little information about what they were guarding.