A third-party application doesn't have the same permissions/protections. Copyright law doesn't always require the author's permission though. If you don't have permission you can still display the content if you have a legal justification. The Fair Use Doctrine is the set of factors to determine whether or not publication without permission is justified.

The transformative nature of the publication is only one factor, although it's the most important. Another factor working in OP's favor is that the use is non-commercial. He says in his blog post that he maintains the site on his own in his spare time, doesn't charge users to download the app, and from what I can tell, doesn't have any ads.

I really think OP could make some small changes and get it approved by the app store.