I'd walk down to the hotel bar and tip the bartender a couple dollars to pop it for me- or go to the gift shop and overpay for a new Disney corkscrew, that lives in the suitcase from now on for next visit.
They did not previously have corkscrews in the studios, but in May when I was there the studio had one. Still needed to go out and buy real wine glasses though as I can't stand the plastic ones. And plastic forks and knives - yuck! Good thing we did not plan on eating in much.
My solution for all this - corkscrew, wine glasses, silverware - was to get an owner's locker. Works great for that and my and DH's golf shoes.