I think the football options are unpalatable enough that UConn can't move to the BE.
Currently the AAC pays more than the BE (granted the BE pays just for BB while the AAC is paying for FB & BB), but if UConn moves to the BE they would be taking a cut in revenue. I don't see that lost revenue being replaced by a football-independent, or by a football-only member of the MAC (assuming the MAC even wants to take UConn as a football-only member). I also don't see dropping to FCS as an option, because the state needs UConn as FBS to recoup the costs of the stadium and because it would preclude UConn from ever moving to a P5 conference like the ACC, which is their ultimate goal.
So the options for UConn are to remain in the AAC, or move to the BE while keeping an FBS football program and giving up revenue. That revenue drain would mean that UConn would continually have less money to support their athletic programs, which could cause all their programs to whither.
I also don't see the benefit for the BE. They are adding a school with an FBS football program, creating many of the issues that the Catholic schools eliminated by splitting from the old BE. Plus, the BE knows that UConn would bolt should the ACC, or another P5 conference ever come calling. They really don't need that instability. And I don't see UConn bringing enough revenue to the BE to make any of that worthwhile. They'll add another mouth to feed, and likely bring in no more revenue than is needed to cover the costs of feeding that extra mouth.