What military document states that berets with nylon headbands are not authorized?

SSG (Join to see) I checked both AR 670-1 and DA PAM 670-1 and neither one mentioned anything about what materials the beret is made of, just how to wear it. I would approach one of those Senior NCO's and ask them where do they found that info since yourself have been looking and can't find it anywhere. If they tell you find it yourself again, I would say that they are full of shit and have no clue what they are talking about.

I've also never seen any documentation on this. When I was in the 82nd I had a beret with a nylon headband. Nobody ever said anything.

The only document that would specify the material is the manufacturer design document, which is what you see in the tag of all military USGI garments, in this case the Beret will be constructed according to military specification #AA-55184.

But, in truth, that's not something most people would reference. You should be fine, the only other thing is if the Commander says he wants all Soldiers to be uniform (the same), and could require a "leather" brim.