My boss' are mounted on a Capita Sierrascope and when I first mounted the bindings, I also thought the screws were pretty short. I was able to get a couple of screws started and I tightened down the bindings. As Leo said, I strapped on the bindings and jumped around and what not, then I tightened again. I also don't trust simply tightening at room temp, I always check back at the mountain since metal contracts with cold. I never had to remove washers, but it sounds like a good idea if you can get the screws to tighten at all. Good luck!
BTW, losing a whole binding sounds pretty crazy!