After a redirect, how can I make the browser show the new URL?
You must force a client-side redirect.
In a Preparation action, use as destination an "ExternalSite" instead of using the screen directly.
The GetEntryURL is a method from the HTTPRequestHandler extension. Just do a Add Remove References to add it. The GetEntryURL method should use another Entry Point, and not a screen directly.
Use the action MakeAbsoluteUrl() on the extension HTTPRequestHandler to garantee that your URL is always right in every environment.