Large responses hang while small ones work

Classic MTU or path MTU discovery failure, usually with a tunnel in the path. Test by lowering the MTU on one side and retrying; if it fixes it, the real problem is ICMP being blocked somewhere, which is what breaks discovery.

Something in the path has an idle timeout, and the fix is keepalives shorter than it. Set TCP keepalive on the socket and, for HTTP, send a periodic ping frame. Raising the timeout on your side alone never helps, because the proxy is the one hanging up.

When you stop noticing it. If it has not surfaced a problem in six months, it is either solved or it was never the bottleneck you thought.