You could create a GITHUB_PAT with very limited scope permissions, perhaps using a dedicated testing account. I think you can turn on PR env var access. If not, you can define it in right in the testing code, possibly with some level of encryption/obfuscation so that it’s not available as plain text to crawlers.
I add a second environment variable, HAS_CARL_PAT="true" or something like that, and have these tests skipped if that is not set. (Checking that it has any PAT might not be robust, since it is common to have a PAT with no permissions just to get past rate limiting).
I also use a separate GitHub Machine account (as GitHub recommends, though you are only permitted one such account) to generate any such tokens, rather than deploy tokens that can access my account.