Add a button to the entry edit screen via plugin
Not sure if you can put it next to the Share button, but check out the Triggers feature of Button Box...


  https://github.com/supercool/buttonbox


You can create a "Triggers" field, which is just a button that hooks up to JS. Then use the JS to ping your Controller via AJAX.

The button exists as a field, so you could just put that field at the top of your field layout. It's not well documented, but it's really easy to do.