I think we'd all agree that playing First base would greatly reduce the risk of injury to Josh.
If so, seeing how good he is when he's healthy, and seeing how 1B isn't exactly a strength for us, doesn't Josh at 1st make sense?
And yes I know, learning 1B isn't easy, but Josh is (A) one of the greatest athletes in the world, and (B) a lefty. I'd bet money he'd be VERY GOOD at first in a relatively short time.
And as to the question of "Would he agree to it?" I personally think we really need to get out of the habit of worrying too much about what a player prefers. He's paid to play baseball, and management is paid to decide where he plays.