@petercooperjr, that would probably catch quite a few of these non-persistent Docker situations!
You know something else that might be useful: an HTTP header that somehow connotes "this resource was moderately expensive to provide to you; please don't make this request in an automated way in a non-persistent environment". Then we would need HTTP clients that run in Docker (etc.) to be able to notice the combination of "I'm non-persistent and automated" and "the server says this is expensive" and generate a warning to the user. (That's quite a few pieces to coordinate, but on the other hand it degrades gracefully because if any piece is missing, the notification just doesn't happen.)
I also imagine that this could be useful for testing environments that spin up a whole virtualized environment and install dependencies or package updates there, then immediately destroy that environment. (I'm not sure what the maintainers of those environments would be willing to do as an alternative in order to have greater persistence, but I bet they can do better than their current naive solutions.)
(like I remember Certbot was using Travis, and it at least gave the appearance of downloading an enormous quantity of dependency packages into a VM to run a fresh test for every single commit on a PR branch!)