How can I make in-app purchases with a different Apple ID than the app was purchased with?
Purchasing from one account and then using another for in-app purchase (IAP) isn't currently possible. 

Since each application is digitally signed when you download it from the App Store - you are tied to using the same account for in-app purchases as for the initial purchase.

If it is a free app, you could back up the device to iCloud, delete the app, re-purchase the app under the correct account and then restore the device from the backup. If you are not worried about saving game state, you can skip the backup and the restore steps.

If the app is not free, then you are faced with deciding to buy the in-app purchase from the original account or re-buying the app on the correct account.