Add a "Save" action in local task to a node
Local tasks are the wrong tool for the job. Local tasks are for providing context-sensitive action links (e.g. Edit, Delete, View) around an item (e.g. node/user). Trying to make a local task become a form submission button would involve all kinds of hacks that probably would be more trouble than its worth.

If the goal is to alter the save button's position for UI concerns, look at using hook_form_alter to alter the appearance/positioning of the node form's Save button. For example, look at a module like Floating Action Buttons that uses a form alter to place the action buttons in a fixed location.