Attach a submit handler on an ajax-y form (hook_media_format_form_prepare_alter)
It always pays to test your assumptions.  The media_format_form only saves metadata edits into the embedded media tag in the edit text contents; the alt and title information attached to the file object are not adjusted.  There is an advantage to this, in that different instances of the same picture embedded in different locations can also have different alt and title text.  Unfortunately, though, it also means that if an image is uploaded via the embedded media image tool in a wysiwyg editor, then the alt and title fields will be empty the next time that image is used, whether it is used as an embedded image or attached to a media image file field.

It would be helpful if media_format_form would save metadata changes back to the file object, just as media_file_edit_modal does.  However, it appears that at the moment, that capability is not present in the code.