Eay-peasy. Someone in the AF is being a Richard Cranium because they think they can.
When I am called to court, or need a document notarized, or anyhing else that requires a response "under oath" I remind the person that I am allowed to make an affirmation and it is just as legall binding.
Here's the current text of the oath on enlistment: http://www.history.army.mil/html/faq/oaths.html . Note that parenthetical "or affirm" - making affirmation negates needing to utter the "So help me God" statement at the end and a short discussion with someone in JAG should clear it up.