Having the right to work in the US doesn't entitle you to a job. And there are plenty of reasons government institutions would want to disallow non-nationals a position. The whole point of citizenship is, after all, buying into the system; if you're not "in" your allegiances or potential recourse can always be argued to be limited. Dual-Citizens, such as myself, are in a similar boat for the same reasons (unless they opt to renounce foreign ties).
I don’t believe this job actually would be possible as the level of clearance requested is usually denied to dual-nationals (without renouncing, of course).