Printing documents such as an invoice requires a set of actions to be run. However, the API doesn't allow for triggering a set of actions.
But I may offer another solution used at another customer - have the process all be in Method rather than ShipStation and QB. At the other customer, the user selects the Sales Orders that he wants to send to ShipStation and clicks a button in Method. This button creates shipping labels in ShipStation for each Sales Order selected, sends them back to Method, and saves them in the SalesOrder table. It also creates invoices and emails the customer.
While it does require a combination of customization and API development, but gets you what you need - both process and 1 system.