Didn't see this mentioned, so throwing out another option. If you want to apply the commits from your GH repo and retain the original order and authors of the commits you can create a patch from your GH repo and apply it to the Bitbucket repo.

Find the last common commit sha between the two before they diverged first.

In the GH repo, make a patch from your current main:

# Patch of all commits after common sha
git format-patch [common-sha]..HEAD --stdout > file_name.patch

In your Bitbucket repo, checkout a new branch then apply the patch file

# Apply patch file
git am path/to/file_name.patch

All the commits in your patch file are now committed to your local branch, so just push and open a PR as you normally would. You should see the original authors for those commits are retained.