They don't have to actually bargain in bad faith, though. They can get to a point where they've exchanged offers and neither side is willing to move.
The mediator can't get them back to the table under your legislation. Why not? Why not have the mediator have special powers to say that the parties actually have to get in a room together and bargain for the next two weeks and to set the schedule for bargaining? If the goal is to actually get agreements, why not empower the special mediator to get people in the room?