So, I've read all the recommendations, accusations, reasoning and can't find where the customer went wrong, unless he
IS in fact, a bold faced liar.
You said he's a nice guy, a new customer and he tried to fix a flaw in your work. I know for a fact, it's sometimes hard to see a small bubble in vinyl, let alone perf. It does look as if there was a problem here, but no one can be sure what created it or who started it. So, if it was there and he handled it wrong, it still needs to be fixed. Your dime in my opinion. If he gouged it and is lying about it, big whoop, like mentioned, about $20 in materials and some time. You now know how to address this customer in the future, if he returns.
Calling someone a liar, when you have no proof of it, is sometimes a death sentence against all future work..... and not just his. Most businesses are built on reputation and if you go around calling people liars and cheats, no one will wanna do business with you regardless of who was at fault for a $20 repair. Is your future and reputation not worth you taking a stride to the side ??
I'd fix it in a heart beat. I just did it yesterday and to a much higher number than $20. Mistakes should be figured into all calculations regardless of who is at fault. The less you don't need to use that, the more you save up for when it does happen. It's like a little insurance plan built into things. We call it the 'SH!T factor. Not to be confused with the PITA factor.
Do as you like, it's just a suggestion from someone whose been around the block a few times.