This just seems needlessly complicated. Performing writes on one endpoint and reading the response on another just seems so wrong to me. An alternative could be that the "client" generates a session id and the start of the chat and make http calls to the server passing that ID in a query string or header. Then, the response is sent back normally instead of just sending 202.
What benefit is SSE providing here? Let the client decide when a session starts/ends by generating IDs and let the server maintain that session internally.