Earlier quoted context omitted.
You don't need websockets, just Connection: Upgrade to anything you want. You can upgrade directly to ssh protocol and just pass on decrypted data from https socket to local port 22 from then on with no further processing.
Proper DPI can tell that wouldn't be acting like a typical HTTP stream, encrypted or not.
Tangential: but I recall reading about a similar technique used on SRTP packets to guess the phonemes being uttered without needing to decrypt the traffic.