You can try and if they say no just go back to your room and get your passport.
We now have digital Driver's Licenses but I've yet to hear of any spot around me that accepts them to buy alcohol. They are used for TSA. We def can not present a photo of an ID. That would be the easiest way to have a fake ID.
They should be very strict because if they get caught not following Florida law then they lose their job and Disney will be fined. And yes, there are often "narcs" testing Disney and other businesses all the time. A CM can be charged as well.
The MB has no information in it, it would have to be attached to your
MDE ... not likely info they want to risk having in their data base. AND it would be too easy to share MB which brings you round to needed an ID to prove that is your MB so pointless.