You can still get these - I bought a couple of packs when I switched to my first set of USTs, Hutchinson was the brand. 10 years old and still pliable, super easy to use.You might be too young to remember these, but back in the day you could get these vulcanized rubber patches that you would actually burn on to a tube. I wish I had one of those I'd apply it to the inside of the tire.
Otherwise yeah I'll try something like that.