So after opening a ticket with Codeship and explaining to them what I was trying to do I got told the following:
"I don't think that we don't officially support the GitHub Container Registry; but, checking with the team on that. You likely can get it to work if it operates in a similar manner as the registries listed in the documentation."
I then tried it and it worked as expected. I also later received an email saying
"The engineering team confirmed that this method should work for you. We are going to update out documentation soon with a proper guide".
In my case I had to create a credentials.env file, add the repository credentials and GHCR url instead of the method mentioned in the question to generate the dockercfg file since I am doing this on Mac OS because the newer versions of Docker have changed to store credentials in the macOS keychain rather than in a configuration file. The instructions can be found here
I hope this is useful to whoever comes across this in future until they have updated the documentation but as of 28th of January 2021, it has not yet been updated.
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