I’m currently using GPP along with WP Offload S3 and WP Smush Pro. I activated the option “Once a file has been copied to S3, remove it from the local server.” But this was after I uploaded the favicon file, so the original copy is still on the server (Siteground), I just checked.
Anyway I noticed after activating this option the favicon is missing on post pages. I don’t have premium for WP Offload S3 so I can’t get support from them. I also know this isn’t caused by GP but I wonder if there is some code (snippet) I could add so it forces the load of the file in the server and not S3.
Also, maybe this could be a feature to consider, allowing GP to use Amazon S3 files for favicons.