Which Tyres?

As a puncture proof tyre is generally going to be thicker then you may be after something that doesn't exist. They tend to use a bit more rubber combined with some extra strips of stuff (kevlar, protex etc,.) so they are not going to be as thin as a race tyre.
You would need a 20c and there are not many of those to start with.

Have you thought about just wheelying everywhere?

Nah, I'm all good now. A 23c Re-fuse will fit, so that's what I have ordered.
Comparing wheels at Easts on Wednesday it looked like most tyres are fine, but Stelvios are just extra-tall.