My understanding is that a BoR must be unanimous in order for the scout to advance.
Well, unless there is a successful appeal. Hopefully this situation can be worked out so there is a unanimous board and no need for an appeal. There is also the possibility (from the original post) that the Scout could have his BOR in a Venture Crew, which as I understand it would require the Scout to transfer his primary registration to the Crew. (We had a Scout who was registered in both the troop and crew, who was considering doing that before starting on his Eagle project-process, due to "issues" with one of the leaders in the troop. He would have had all of his Eagle project signoffs, Eagle application, unit leader conference and BOR all with the leaders of the crew. Fortunately cooler heads prevailed and the Scout never did transfer and the Eagle process was completed within the troop. And they all lived happily ever after. Well, at least they all tolerated the situation and got through it with nobody yelling at anybody or going to council or switching units.)