I am running this on the primary, the question is if i can run this same script on the replica.
You can based on what you told me about the snapshot; however, the pg_start_backup() command will fail.
Note: What you are doing today most would consider it a “proper” online backup; when you start running the script on the replica then you will end up with a “crashed” backup. The “crashed” backup only works as you can guarantee a point in time atomic view of all the data files using a single atomic snapshot.
You need to remove the pg_start_backup() and pg_stop_backup() commands from the script. The backup should occur faster as it is no longer waiting on pg_start_backup() to issue checkpoint; however, that also means that recovery times increase for the same reason.