We are trying nail varnish for my 6 year-old daughter. Like duct tape (which we could never get to stay on either), it deprives the verruca of air, and hopefully kills it.
We have tried everything else (including duct tape, banana, bazuka and home freezing - which provoked the loudest screams ever). And so thought that nail varnish would be easy and stress free. I just paint it on and every few days in the bath, we put a polythene bag over her hand and she tries to peel the varnish off, along with any bits of verruca that might come too. If it doesn't come off easily then I just paint some more on.
Like your DS, she won't let me touch it in any way, to peel stuff off or file it. (I tried letting her file it herself for a while, but you have never seen less effective filing in your whole life!) Hence her peeling it off herself.
We've just been trying nail varnish for a week or so, and something seems to be happening. It's getting slightly smaller I think, and we have got down to some black strands (I'm hoping this is a good sign) and they're sticking to the nail varnish, meaning that they'll be pulled off next time. (Sorry for the descriptiveness, it's not attractive).
I'm going to stick with this method, as by covering it it should hopefully not spread (although I am paranoid, so using a plaster too). Plus it seems to be improving and isn't as stressful as lots of other methods. I think it will still be a long road, but with this method it's not getting worse at least And if verrucas do just go away on their own when they are ready, this is unlikely to interfere with this process.