After some experimenting by Delicious Brains (producer of the "offload S3" plug-in) on a staging site I created on WPEngine, it appears that WPSmush Pro is causing the WP media manager to respond slowly when attempting to put content on pages, as well as when adding content to the media manager. I have copied in the text of Delicious Brains' email to me. We have validated their finding by deactivating WPSmush Pro and witnessing a significant performance increase. Kindly advise...
Thanks for setting up and giving us access to the staging site.
I had a little play, and sure enough it is pretty slow when showing the Media Library in either list or grid mode.
However, on a hunch I temporarily deactivated the WP Smush Pro plugin, that sped things up immensely. When I activated it again the slow Media Library returned. You may want to contact the makers of that plugin.
On a side note, you might get a little more performance out of WP Offload S3 by setting up a CDN such as CloudFront to pull from the bucket and switch to using that in WP Offload S3.
We have a setup guide for CloudFront (slightly out of date with current UI, will be fixed soon) https://deliciousbrains.com/wp-offload-s3/doc/cloudfront-setup/
Please get in touch if there is anything else I can help you with.