Many apps don't work for secondary accounts
As far as Android SDK documentation goes:

    As an app developer, there’s nothing different you need to do in order for your app 
to work properly with multiple users on a single device. 
...

    To be certain that your app behaves properly in a multi-user environment, do not refer 
to your internal app directory or external storage location using hard-coded paths and 
instead always use the appropriate APIs:


Hence theoretically speaking, if the app developer had used appropriate methods this app should work normally. In that sense, I do not think you have any better way other than reporting this to the app developer.

Rooting your Nexus 7 doesn't going to solve this issue - in my opinion.



Update: I sent an email to the support team of the app developer quoting this question. I received a reply as:

Thank you for your information.    
We have filed the issue with engineering team.    
We'll try our best to fix them ASAP.


which in my opinion supports my assumption.