My opinion is that Japan and the U.S. should do their own project, and let the Europeans do whatever they want.
As for funding ideas, we could consider redirecting the money spent on funding the United Nations each year to this project.
If we could convince Japan on that score (not likely unless they are rebuffed on their quest for a Security Council seat, and even that probably would not be enough) that would remove a large portion of the U.N.'s funding. I haven't checked, but I believe that Japan and the U.S. together contribute in the neighborhood of 40% or so of the total budget for the U.N.
Japan pulled about a third of it's funding from the UN earlier this year.