It would seem that inside his shirt would be a bit risky --- they could be visible, inside his waistband would not prevent them from slipping or being wrinkled --- to me the most logical place you'd put documents you were removing and couldn't be caught removing would be in your underwear which would help hold them in place.
Exactly... and yet Burglar claims not to have noticed. Presumably he drove or was driven home, but wasn't the least bit uncomfortable while sitting. And this clown was National Security Advisor? God help us!