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This appears to be a similar issue to.
I'm using the latest dev versions of both File entity and Media. I have a multiple value File field using the Media file selector. When I click 'Add media', choose from the modal, and submit, the 'Add media' and 'Delete media' buttons output, but no 'Edit media'.
If I click 'Add another item', a new row is added and the 'Edit media' button missing on the previous row outputs.
If I comment out the #access property in media_element_process() the 'Edit media' button outputs correctly, so it seems like it's a combination of permissions and AJAX.
I've attached screenshots to illustrate the issue.