Ideally, you should do this through a module, but this does bring a lot of detail that is not the focus right now.
Notice that the Block you are trying to change is part of the core and is in:
app / code / core / Mage / Adminhtml / Block / Catalog / Product / Edit.php
There in the constructor runs: $ this-> setTemplate ('catalog / product / edit.phtml');
I think this is the view you want to manipulate.
There for the button action, you would direct to a controller of your module, where the prefix would be configured in the config.xml of it. For more details, see the sources of Magento modules that are in GitHub.
I will indicate 2 that I had some contact, where in the first I put a reversal button in the order details administrative page: