There's an internal contradiction here.Never underestimate budget scrooges. We are only into the early stages and if cost overruns occur they may try to cut this one back.
A "budget Scrooge" wouldn't spend anything.
But then you mention "cost overruns," which means they are indeed spending quite a lot.
Disney does indeed usually go overbudget. That means they spend more than they wanted and more than other theme parks do. But it does mean that if one part of the project is running over, it gets scaled back, or, some other project gets cut.
But they still (over) spend the money. Not hoard it like Scrooge.