Chris... Sounds like you're missing the backing plate that goes with the cleat. It looks like this:
Unless the washers are the right size, it's easy for the rivet to pull through. Depending on the application, our loft uses alum pin rivets, brass pin rivets or stainless hardware. I'm looking at a sail that has alum pin rivets in it. Pin rivets are different than the pop rivets used throughout the boat. It takes a special tool to set these. Likes kind of like this:
If you've got the backing plate and still can't get it to hold, then I'm missing something. The loft guys can probably fix in a few minutes if you can't get it to hold.
p.s. Sorry for the delay. Got busy here for awhile.