I haven’t personally used the S3FullAccess role but it would seem like it should give well, full access.
You’d want to make sure you’re using the same/correct credentials and so on in the Omeka config as you used successfully elsewhere, but I imagine you’ve checked that already.
S3 has some options to disallow “public” things in buckets… have you checked if you have those enabled maybe? Under “Permissions” when editing a bucket there’s a setting called " Block public access to buckets and objects granted through new access control lists (ACLs)" that could be a problem if it were enabled.
Omeka Classic’s code for dealing with S3 is pretty old: you might need to change Object Ownership back to its default of “Object writer,” since Omeka does try to send an ACL when it pushes files to S3.
There are also more detailed settings under “Block All Public Access” and you can check those; the one I mentioned is one of them.
The built-into-Omeka code will work fine, though you could move to one of the others that’s in a plugin if you want. Typically we see people use those if they need to support a “newer” region that requires newer code.
I’ll also look at adding an option to the built-in code to disable setting per-object ACLs to avoid this kind of problem cropping up in the future.