There isn't much overhead here other than connection setup. For HTTP/1 the connection is just "upgraded" to websockets. For HTTP/2 I think the HTTP layer still lives on a bit so that you can use connection multiplexing (which maybe be overhead if you have no use for it here) but that is still a very thin layer.
So I think the question isn't so much HTTP overhead but WebSocket overhead. WebSockets add a bit of message framing and whatnot that may be overhead if you don't need it.
In 99% of applications if you need encryption, authentication and message framing you would be hard-pressed to find a significantly more efficient option.