Wishy that sounds like it would work but how would I test if its a new install or not?
the best way in my opinion (requires some knowledge in with AJAX plugin, also MySQL and a little PHP is needed) is, building up your own little framework/api where a user has the option to register (not speaking of Facebook - or something you dont want to have ; Username & PW is totally enough.. maybe an optional mail-address for beeing able to recover his password). This enables such great ways and features to extend an app (having the same savegame on different devices, etc.).
I think in any way you should head for some webstored data (speaking of MySQL, PHP-to-TXT-Scripts, etc.). When I read "Restore purchases" I understand "restore them under any circumsances", meaning: even when flashing a new ROM to your device or changing it (after a guarantee repair, for example), I am able to restore them.
An another way could be including Google Play services into your project (there are plenty tutorials in the forums/tutorial-section for it) - but even then, if you want to cover all scenarios of loosing data, you have to store things externally of a device (where MySQL comes back again ;D).
Good luck with your project and have a great weekend,