In my case with this load balancer, I think it's just badly written. I think it is set to hold ALL data until the server ends the connection. I have tried leaving my SSE open to send over a few megabytes worth of data and the load balancer never forwarded it at all until I commanded the server to close the connection.
The dev who wrote that code probably didn't think too much about memory efficiency of proxying HTTP connections or case of streaming HTTP connections like SSE.
I think there is a HTTP/2 flush instruction, but no load balancer is obligated to handle it and your SSE library might not be flushing anyway.