I honestly would prefer what you are suggesting, even if that meant using iframe on the original domain to host the AMP cache from Google.
1. example.com/long-url with content and meta tag specifying AMP url (2) below.
2. example.com/long-url/amp that embeds a full-height 0-margin iframe to google.com/amp/example.com/long-url and has no external styles/JS.
3. Google caches content from (1) example.com/long-url and when user clicks it, directs them to (2) example.com/long-url/amp.
Other than 1 extra HTML page, everything is from Google’s AMP cache. The browsers do not need to lie. And if Cloudflare hosts the (2) AMP page, it can be fast as the rest of their services.