I'm not going to argue that it was an ancient symbol, nor that it was used in 1066, but if we're drawing coats of arms, the only sure coat of arms we have of the Byzantine Emperor is the eagle, and then the tetragrammatical cross. Are they late? Sure, but the attributed arms of the Heptarchy are made up, we have no idea what real symbol half the world used, and in any case William the Conqueror should definitely not be sporting three lions passant.
By 1270, as the Palaeologues became Emperors, the Tetragrammaton was quickly seen as the symbol of the Empire from abroad. The Wignberger Roll, dated 1270, shows this, and most armorials show the tetrasgrammaton more often than they show an eagle
If the game doesn't have any later dates than 1066, then I'd side with you, but for a whole 1066-1453 run, I'd go with the tetragrammaton. I don't mind using the Chi Rho, but if it comes within a coat-of-arms frame, I'd rather see a coat of arms.