Your setup sounds really intersting, do you have any links for tutorials on how to setup something similar?
You can also "cheat" and instead of registering your VS Code server instance with Microsoft and find the path in your ~/.vscode-server or whatever directory, and there should be a shell script that starts it and runs VS Code on port 8000. You can then set up any reverse proxy/TLS/auth layer in front of that and access it similarly.
I think the VS Online (now called VS Codespaces) experience is a bit better but the process of creating an account and logging in is messier.