Just to be clear, Did you bought the paid, complete version on the store?
If is that the case, they are treated like two different apps, so you can make a backup file on the free version and restore it on paid version.
The thing is that you could have paid for the full game on the option "store" that you see on the main menu, so you don't need transfer anything, it just unlocks the rest of the game.
Good hunting...