How to re-download in-app purchase
To answer your question 'Are there any alternatives to get the mighty eagle back?' Unfortunately not. It's up to app developers to support redownloading of in app content, and it certainly seems that you have found a bug. 

Get in contact with Rovio at http://www.rovio.com/en/about-us/contact  or contact@rovio.com and explain your issue - they will be able to help you further