Just download the book to your PC like it wants you to do. Then forget about that downloaded book... it will only work on a Kindle device anyway.
Now open Kindle for PC and go to your Archive and locate the library book. Double-click it... and when it's done downloading, start reading it.
Granted... I
do have a Kindle device, but my library loans have always been available in my Kindle For PC archive. And I've downloaded several to the app that way.