Bf is a separate state agency. When the program was first started I believe it did cover tuition at all state schools, but tuition was a lot lower (about $3000) and it did not cover fees. As tuition was raised, BF continued to cover about $3200 of tuition when tuition was about $6500. In 2016(?) BF again raise the level back to cover tuition at the top award, fees, and a $600 book stipend. It covers 75% of tuition at the medallion level, but no book stipend.
UF can raise tuition without BF covering it all; it’s not unprecedented. The BF agency could word it that BF covers tuition up to $XXX or whatever they decide. BF pays a per credit fee to private schools too. The amount is 120 crdits x $213.55 =$25,626 (about 4 years of tuition at a state school) (and they really do cut you off when you hit that 120th credit - ask how I know). I think right now they pay up to 18 credits per semester at public schools, but the student would have to pay anything over that. It used to not pay for summer tuition but it currently does.
Another consideration for Florida is that a LOT of people have prepaid tuition plans which they bought long ago and those plans could be for 4 years of tuition and fees. If tuition rises substantially, those plans could be in trouble. I do think the plans are run by a private company, so Sasse may not care if raising tuition eats into profits.
But I think Florida needs to raise tuition. And I think combining some of those 200 department could cut down on admin costs.