The other answers do not seem to touch on a key aspect here, and that is you make no mention of requiring supporting a web browser as a client. Most of the limitations of plain HTTP above are assuming you would be working with browser/ JS implementations.
The HTTP protocol is fully capable of full-duplex communication; it is legal to have a client perform a POST with a chunked encoding transfer, and a server to return a response with a chunked-encoding body. This would remove the header overhead to just at init time.
So if all you're looking for is full-duplex, control both client and server, and are not interested in extra framing/features of WebSockets, then I would argue that HTTP is a simpler approach with lower latency/CPU (although the latency would really only differ in microseconds or less for either).