For me, the best coffee at Disney World is at Tea Traders by Joffrey's in The Landing area of Disney Springs. Back in May, they had a new menu with 4 or 5 different coffees from various parts of the world. Each cup was made on demand. I honestly can't recall if it was French Press or pour-over. But, it was far superior to any other "quick serve" option on property.
(I recognize this isn't helpful to madmar because Disney Springs isn't near Disney's Wilderness Lodge.)