I bought a pair of Continental "puncture proof" tyres for my bike recently. I had an almighty battle to fit them, which resulted in a puncture.
Once I'd fixed the puncture I realised I had
to get the tyre on by hand (which is what I would normally do anyway). A bit more fighting and it finally went on with the tube completely deflated- normally I put some air into tubes when fitting to avoid it twisting or getting trapped under the bead.
Try a var tyre lever. Excellent tool.
Var tyre lever does look like the answer. Thanks for suggesting that, I was quite worried about how I would cope if the puncture proof tyres didn't do their job.
1979 Dawes Fox
1982 Dawes Super Galaxy