Fabric won't work for this application as it could get wet and, um, icky.
I just had another thought, though. If I could put some form of rubber thing underneath one of the toilet seat standoffs, something that would guarantee me a difference in clearance when weight is applied vs not, I could then design a mechanical switch to work within that clearance difference and not need a pressure switch at all. Then the question is what can I put under a toilet seat standoff that would give me that difference? A small rubber ball would raise the seat too much.
Another thought - a switch behind the lid. It would require someone to lean back, but that might work and I could use a basic momentary contact switch for it.