From the discussion I don't think a code share is involved. I believe the OP has purchased an Interline ticket from an AUH airline that includes a sector on FlyBe.
In this circumstance each airlines T&Cs apply. However a careful reading will show that both of them apply IATAs " most significant carrier" rule which says that the baggage rules of the MSC applies to the Interline journey. So Air AUH's rule applies. IATA no longer fixes baggage charges, the MSC's charges apply.
In the event of a code share, your contract is with the coded airline, not the operating airline. This is not an industry rule, it is a basic tenant of contract law. Not all airlines like this, but it is the view of the judiciary in every jurisdiction that I know.
Not all airlines participate in the MSC, but most network airlines do.