How much extra costs did he want to shift to the Federal government?
No one knows, not even Gov. Christie. That's the thing about cost overruns.
The deal that was originally cut is that out of the umpteen zillion dollars -- actually, about eight-and-a-half billion -- that this project was supposed to cost, the fed would kick in $3B, the Port Authority would kick in another $3B, and the New Jersey would pay "the rest," originally estimated at two and a half billion dollars, but Cheops' law says that nothing gets built on time or under budget.
That's actually a pretty good deal; New Jersey gets the lion's share of the benefits for the fox's share of the costs, if it can manage the project with any reasonable degree of efficiency. if it can't manage it, then it has to clean up its own mess.
Turn the deal around, and you can see what a sucker's deal it would be if the Fed had agreed to cover overruns. New Jersey would be managing a project knowing that whatever money they spent would be reimbursed -- which means they could put every politically connected hack on the payroll as a "consultant" at exorbitant salaries, knowing it wouldn't cost them a dime. It would be an open license to pick the Department of Transportation's pocket. A blank check.
Even so, the Fed was willing to pick up a larger share of the cost; it offered to increase its direct funding, and also offered to provide loan guarantees for almost any amount. The one thing it wasn't willing to do was provide that blank check. Christie turned it down because he wasn't willing to take responsibility for his own money, only for other people's.....