This isn't so much a legal strategy but I'd tell him you don't feel comfortable with your lease being lost somewhere, which can be used by people for all sorts of purposes in proving residency, and insist on helping him find it by retracing his steps. Tell him you need the name of the copy shop so you can ask if he left it in a machine or if they found it in the parking lot. Obviously he didn't really lose it but maybe he'll miraculously find it if you make a big enough deal. And if you do end up signing anything, either make sure it's NCR paper so you get your copy right away, or go with him to make a copy.