Error trying to set my site up with Cloudflare
I'm afraid you won't be able to do that. Cloudflare allows for registering root domains, you can't do that with a subdomain. However, you can buy some domain, register it on Cloudflare, point A/AAAA entries to a Azure site you have and change the webserver config to listen to this domain.
In that way you'd be able to use Cloudflare proxy for sure.