Do you have an idea, what needs to be fixed in the extension, so that I could test it?
Unfortunately I haven't worked with GitHub Codespaces yet.

No idea, unfortunately the extension is closed source I believe.

https://docs.github.com/en/codespaces/setting-up-your-project-for-codespaces/configuring-dev-containers/setting-a-minimum-specification-for-codespace-machines

Can I trick code-server into not being displayed as a remote session to plugins or in general?

I suspect it would be a significant architectural change. The problem, if I understand correctly, is that the web client itself uses remote APIs to connect to the server, and the plugin itself also uses remote APIs, but the client is designed to only hold one remote connection at a time. So that would have to be completely refactored in some way to allow chaining remote connections together (client > server > dev container).

Or, the extension would have to be converted into a web extension so it can run fully in the browser without needing a separate remote server connection, but I am not sure how much work that would be (there is a feature request for this here: microsoft/vscode-remote-release#9059).

I tried the plugin on Codespaces and I get this error:

The 'Dev Containers' extension is not available in Visual Studio Code for the Web. Click 'More Information' to learn more.

You could try asking the VS Code folks (https://github.com/microsoft/vscode or https://github.com/Microsoft/vscode-remote-release/) if it is possible to get the extension working on Codespaces or vscode.dev. Any advice they give should work on code-server as well. I think it may not be possible, but I would not consider myself an expert on VS Code.