My first idea would be to modify the Wordpress login function to set the login cookie not only for the Wordpress installation domain but also the InvoicePlane domain. Inside InvoicePlane you would need to add a key or something that may be securely stored in the cookie which will be checked by InvoicePlane.
application/modules/sessions/models/Mdl_sessions.php handles the login authentication so you may take a look in how the required data looks like.
Session validation is done inside the /application/core/User_Controller.php file.