Flight Crew View (and nearly all apps) support carrier billing. This is actually not a setting that developers have any control over (even though some carriers' troubleshooting guides seems to indicate that we do). It is an option developed by Google and Apple for your convenience. All that happens for in-app purchases is make a call to the Play Store or iTunes Store with the item for purchase. Google and Apple handles everything after that.
For Android users, double check your payment methods in the Play Store. Go to the menu in the Play Store app on your mobile device and select Account and then go to Payment Methods. Double check that "Bill my [carrier] US Account" is listed as a payment method. If it's not, then try to add it. If you can't add it there, then it is an issue with either Google or or your carrier.
Apple Users should also check their payment methods in the iTunes Store.
If it is an option there, but the app is not listing it as a payment option, then make sure that you haven't exceeded any spending limits or have changed any settings in your carrier's account that would restrict premium content purchases. There are other restrictions that apply that also might be affecting your ability to use Carrier Billing (for instance, maybe your phone thinks you are still in a different country from a recent overnight, etc).
Google Support: https://support.google.com/googleplay/answer/2651410?hl=en
T-Mobile Support : https://support.t-mobile.com/docs/DOC-3852
Verizon Support: http://www.verizonwireless.com/support/google-carrier-billing-faqs/