Can non iPhone iOS devices (iPad specifically) be used to dial short USSD codes for carrier information?
Nope, not possible! The USSD code would be sent over a voice call of sorts, but there's no phone app on the iPad. The cellular access is strictly for data (at least without jailbreaking).

You'll just have to find a different way of checking your data balance! You could just keep track manually by going to Settings -> General -> Usage -> Cellular Usage. That will allow you to reset your usage statistics and then keep tabs on how much data you've used up to that point.