Getting the full game on a new phone, or after reinstalling

The full game belongs to the Google account that bought it, not to the phone. Install Siegebreaker on another phone, or reinstall it on the same one, while signed in to that account, and the full game unlocks by itself the next time the game can reach Google Play. There is nothing to buy again and nothing to restore by hand.

Once it knows, the game remembers, so the full game keeps working without an internet connection.

Your campaigns and records are a different matter: they are kept on the phone, not with the purchase. See Keeping your progress when you change phones.