This is a terrible idea in so many ways that I don't even know where to start. The Army-Navy game needs no "counterbalance" and the fact you think it does is very strange. I suspect the founding fathers would not only not be "rolling over in their graves" over the Army-Navy game, many of them would probably be in attendance.
You are suggesting an annual game on a date that has for decades been reserved for a storied rivalry between two outstanding military institutions. And you are suggesting BYU do that with a team that has historic rivalries with USC, Navy, Stanford, Michigan, Boston College, Purdue, Michigan State, Pittsburgh, and Northwestern and would probably consider 25 other institutions before it even thought of BYU.
Total pipe dream and the antipathy towards a great military tradition is misplaced.
This is a Greenrake level idea.
But if Notre Dame ever wanted to do it on Championship Weekend, I'd be game. But they won't.