I wonder what gave them the confidence that it would work?

That's an interesting question. My initial guess (short form) was : math.

But after some more research, it seems the basic premise had been well tested in the lab, and was considered simplicity itself, unable to fail. And fissible material was in short supply (after all, it's not like every Tom, Dick and Mahmoud was making it by the bushel back then).

If it hadn't been obvious before, the Project must have made it abundantly clear to everyone that brainiacs were going to inherit whatever was left of the earth after they unleashed nuclear fission. If one of them said stand back, the prudent thing would have been to clear the room, and apparently it'll work was all they needed to say to end the war. Those guys were walking on water.