I think this is one of the big reasons we should've signed a veteran QB in the off-season. Just to manage the game while Rodgers is hurting.
As to keep Rodgers out right now, we can't really do that. Flynn is going to struggle horribly out there and Brohm, well.... He's certainly not ready yet.
Ofcourse, if this is a serious injuy that will be worsened or something, he should sit it out, but if it's just 'playing through the pain', we have a better chance with him at QB.