The usual latency should be on the order of 30-40ms round trip.
Suddenly getting a 500ms latency every minute
No one said anything about that. The client should have some form of dead reckoning, and there should be an edge server that maintains connections. With the right kind of game mechanics, no one notices the server disappearing then reappearing behind the scenes. I cite 250ms as a worst case. It should be possible to get the impact as low as the server skipping one frame every few minutes. The clients don't need to skip any frames.
That's going to be a very expensive edge server!
I'll always grant that fast twitch FPS will always have a problem with this. That's not the genre I'm interested in creating games for, and it's not what my server framework is designed to support. (It is real-time action, but the pacing is more akin to RTS than FPS.)
I would like to see a framework, where the backend of a cloud system can migrate opposite to the direction of orbit, so the same virtual cloud server could remain (by 'hops' where it serializes a copy of itself, then fails over to the copy) more or less geostationary. Such migrations would happen as seldom as once per 10 minutes.
My game server cluster isn't too far away from having such a capability. (I was planning a variety of capabilities, such as using cheap GCP "ephemeral" instances by quickly failing over to such copies.)