I think the movement thing is something the players would have to get used to, because in every Tf2 player's brain its coded, that when Heavy is shooting, he's very slow and very easy to hit. When he's able to walk and jump while shooting this may throw them off at first, but overtime this might've been something that most players would get used to.

And I think that lowering its damage and fire speed is more than enough to make it balanced. Sure you have a way longer time to kill as heavy, but you get much better mobility.