You can now use a relay API in the Claude desktop app.
Anthropic quietly added official documentation for this configuration about two days ago.

Here's how to set it up:
Download the latest Claude desktop app — older versions won't work.
Open the app, but don't sign in yet.
Click the menu in the top-right corner, then go to Help → Troubleshooting → Enable developer mode.

Once enabled, a new "developer" option appears in the top-right Claude menu. Find "Configure third-party inference."
Enter your relay service's Base URL and API Key, then select "Apply locally." That's it.