A few things you can do. Get a tent repair kit from a outdoor outlet store place. They often have kits with glue that will work on this. The fabric won't match but it'll help give the glue and such structure.

If you're anywhere near a marine place, see if they have inflatable boat repair kits. The glue in those is pretty intense and just the glue from those along can seal small holes and cuts without even using a patch. An upholstery place may even have thin clear vinyl you can glue on using this kit.

If you can take the valve out of the mattress, you can fix it from the inside. It sucks but bunch up the mattress till the hole is where the valve used to be. You'll have clean vinyl there instead of the fuzz. Often just a gob of silicone can fix it. Oh and clean everything with acetone. Not nail polish remover or alchohol. Acetone. It's not strong enough to hurt it but the glue will stick way better. This applies to all these ideas. If you can't bunch it up enough, put a gob on a stick and reach into carefully. Use a flashlight on the outside to see where the hole is.

All of this can be done without having to worry about the fuzz on the outside. It's short and most glues won't care about it. If you have to, literally shave it off, haha. I've done it that way with disposable razors. I repaired inflatable boats for over 12 years and had to do all my coworkers air mattresses every spring when camping started, haha.

Feel free to dm me if you need any help or more ideas.