Unfortunately AA is now passing through BA fuel surcharges, even if it's an AA flight number on BA metal. The only way to avoid BA fuel fines is to avoid BA.

Cathay now adds a fuel surcharge of US$124.50 for flights between HKG and Europe, Africa, Oz/NZ, India/Middle East and the Americas, and a lesser amount ($29.50) for southeast and east Asia. Here's their listing.

To see BA's fuel surcharge amounts, you have to go to their Japanese website (it seems they don't want to make it easy to find the amounts elsewhere) but here it is, e.g. GBP179.50 for 9+ hour flights in business or first.

To reiterate, these are not taxes. The technical term for them is "profit." Note you will pay BA fuel surcharges even on mileage redemption tickets, including those booked with AA or other airlines' miles/points.