You need the web site to use a certificate from the same root authority as your client certificate. Otherwise browsers won't present the certificate to the server. That means either warnings on connect or adding the root cert.
I do think if you are doing it with them in person it is doable to add it.
I checked and this does a similar thing. You can use WG to punch out a connection, and I have before. The linked program uses quic in a similar way to WG but doesn't do any of the routing, just registers the connection to a hostname for application level forwarding.
It looks more that it provides an easy setup for an alternative.
Both methods can work on cg-nat as well since you are punching out.