I stumbled across the answer to this question on a hunch so posting the solution in case anyone else has the same issue.

In GitHub, TFLint relies on the GITHUB_TOKEN automatically generated by the workflow to authenticate with the GitHub API.

So in an attempt to replicate this, I created a new GitHub PAT, and added it to my GitLab pipeline as a variable named GITHUB_TOKEN. Then in the pipeline script, before I call TFLint I added the following line:

GITHUB_TOKEN=${GITHUB_TOKEN}