Not the constitution, but security clearance.
Please read the DoD’s stance on security clearance with dual citizenships.
http://www.dod.mil/dodgc/doha/isp.html
Actually, the DoD is my only hope for this mess and if there is something there, they will uncover it while researching for the level of security clearance he needs.
The Constitution is also silent on security clearances. Bill Clinton couldn’t have gotten a high enough security clearance to be his own janitor.