Thanks for the request.
In general, the license is bound to your Google account so it should restore just fine if you configured the device with the same Google account. Please make sure that Google Play and the Google Play services are properly set up on your device.
After that, just hit the "Buy pro version" button and choose Google Play. The dialog should just disappear and the pro version should be enabled again. If you get a Google Play styled dialog asking you for a payment instead, please let me know.