You have the minimum stuff: access to Asset Store (billing services) and Game Center. The game is fairly simple so the architecture to sustain it should too.
The solution must not require setting up a third party server account, nor depend on iCloud.
Intuitively I'd say it's nigh impossible but maybe I missed something.
Data to be saved :
1. Virtual currency (VC), a given amount, bought with several options of IAP (small spending, medium, big, etc.).
2. Some virtual goods (VG) bought with VC (not IAP!).
The VG are non-consumables and can be upgraded too (think of weapons).
How would you proceed to allow such data to be saved so it could be accessed and restored on another device?
If there is really no quickly available option, should we strongly encourage the player to use an iCloud account to save this data (I currently am able to save keys that point to string data)?