Currently, the print sand save functions cannot be triggered programmatically.
However, you can set up the save functionality to autosave at a given interval. Additionally, you can turn on "focus polling" so that a save occurs when the viewer loses focus.
With focus polling, adding your own save button is unnecessary since simply by moving the focus to your button would cause the save to occur. What you might want to do instead is add your "Save" button but make it do something with the saved PDF rather than just update the content in the viewer.