Aha, I got it. I correctly set up the remote url, and the issue was the authentication. After clone using fedpkg clone --anonymous and comparing the .git/config file with the one of a normal clone, I found that the only difference was the credential helper.
I initially overlooked the official documentation and attempted to fork, modify, commit and push code using standard git. As a result, I faced persistent authentication failures when trying to push my commits to my fork on src.fedoraproject.org, even after correctly configuring my API tokens and SSH keys.
However, after inserting the snippet you provided into path/to/my_repo/.git/config, I was finally able to successfully run git push --set-upstream origin name_of_branch.
For others who might encounter similar issues, running git remote -v on my local cloned fork returned:
Once the snippet was inserted, the authentication process involved signing into my Fedora account via a pop-up browser window, after which (I assume) a token was shared with the git instance to facilitate authentication for pushing to my fork.