The ACF gallery field uses the native WP media popup to display the field data, then it uses CSS to hide the usual main / side bar layout and only show the sidebar.
The issue may be that the WP popup contains a bug in regards to user permissions, so can you open up the normal WP popup (via the_content WYSIWYG) and try to edit the image in question using different user accounts?
Does the WP popup allow you to edit the image?
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