What does HackerNews think of wstunnel?
Tunneling over websocket protocol - Static binary available
To keep your wireguard setup more as-is, you could try https://kirill888.github.io/notes/wireguard-via-websocket/ to tunnel that via a web server. In fact https://github.com/erebe/wstunnel which that uses could be used just as well with any other UDP based VPN.
I once tinkered with https://github.com/yarrick/iodine and successfully connected to resources over the wireless on a train, bypassing its traffic capture and sign-up requirement, so that might be an option, though I think fully blocking external DNS is more common now so this is less likely to work²³.
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[0] practically only HTTP(S) permitted, not even SSH, DPI in use that detected just using SSH or OpenVPN over port 443
[1] NOTE: be careful breaching restrictions like this, you are at risk of an insta-sacking if discovered, or worse if operating in some securiry environments!
[2] and the latency when it does work is significant!
[3] and that much traffic over port 53 might get noticed by the heuristics of data exfiltration scanner, encouraging sysadmins to notice and implement a way to block it
https://github.com/erebe/wstunnel (linux + mac + windows)
https://github.com/erebe/wstunnel (linux + mac + windows)