You don’t have to fill out FAFSA to get BF (and as I said Benac is the same agency) but the payment does go through the FA offices at the colleges, and the colleges can make you prove citizenship/legal status before releasing any aid.
We did fill out the FAFSA, so that wasn’t an issue for us, but for those who don’t the schools could have another form, like UF has the form I linked in post #4