When programmatically uploading a file to Drop Off Library creating by Content Organizer feature, all pre-defined rules won't be automatically executed, and the document won't route to its destination right away even you have the proper rules set up. I thinks the only way to triggle the routing is by UI.
Reduce the interval of Content Organizer Processing timer job for the web application. By default the timer job is scheduled to run daily. You could reduce it to run in every minute.