Hi @brett-polivka,
I haven't tried ghcr, but in my experience GitHub is really unstable on the API/integration side when it's outside of their core source/Git hosting functions (e.g. Packages are notoriously buggy), so if your PAT is okay, then this is the most likely scenario.
The Connector Health Check for Docker uses the catalog API (/v2/_catalog), and the response should look something like this: https://proget.inedo.com/v2/_catalog. This endpoint is particularly buggy in other repositories (especially ones that require authentication/authorization), so my guess is GitHub introduced a regression or something on their implementation.
Another possibility... 403 is access related, so it could be something on your proxy side; please also check your Proxy settings, or something on your network side.