The way the current architecture works—as far as I know—is your assumed "server caches the generated output" step doesn't exist. What you get in your output is streamed directly from the LLM to your client. Which is, in theory, the most efficient way to do it.
That's why LLM outputs that get cut off mid-stream require the end user click the "retry" button and not the, "re-send me that last output" button (which doesn't exist).
I would imagine that a simpler approach would be to simply make the last prompt idempotent... Which would require caching on their servers; something that supposedly isn't happening right now. That way, if the user re-sends the last prompt the server just responds with the same exact output it just generated. Except LLMs often make mistakes and hallucinate things... So re-sending the last prompt and hoping for a better output isn't an uncommon thing.
Soooo... Back to my suggested workaround in my other comment: Pub/sub over WebSockets :D