I had a similar issue at one point but if I remember correctly I just had to have my webserver send the header section without closing the connection.
Usually things would just get streamed through but for some reason until the full header was sent the proxy didn't forward and didn't acknowledge the connection.
Not saying that is your issue but definitely was mine.
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.
Usually things would just get streamed through but for some reason until the full header was sent the proxy didn't forward and didn't acknowledge the connection.
Not saying that is your issue but definitely was mine.