Actually, most of the hardware stores DO carry butyl tape just not where you might think it is. Look for a 3-6' long roll, 1-2" wide, in the PLUMBING section or the ELECTRICAL section. I've bought 3M's prime quality "butyl self vulcanizing tape" in Home Depot, right next to the other 3M electrical tapes, it is used to waterproof cable and antenna splices. These tend to harden up over time, not rock hard but layers fuse. Regular butyl "caulking" tape will stay gummier longer.
Sometimes they stock clear silicone tape--which is pretty much the same product, doesn't quite self-fuse as permanently but can be used as a gasket material just as well. Sold in the plumbing sections (along with butyl) as a hose/pipe leak patch material, too.
I'm assuming the tape is going to be laid down flat as a gasket under the hinge panel--not stuffed in the screwhole. Either of these should work as a gasket material, without needing to buy a huge roll from a special place.