Yes, websockets should be fine. Making my proxy support websockets was a little difficult, but I think it works now. If the traffic is compliant with the proto

Yes, websockets should be fine. Making my proxy support websockets was a little difficult, but I think it works now. If the traffic is compliant with the protocols that ports 80 and 443 are supposed to use, it's not a problem. The one that really gets me is a thermostat using a proprietary protocol over port 443. This protocol is one where the server sends the first data over the connection; in TLS the client always sends first. So my proxy was just waiting for the TLS client hello while the thermostat was waiting for its server's message. If the thermostat had sent something first, the proxy could have seen that it was invalid TLS and passed it through; instead it deadlocked.