Clearly shows under the HKG-DOH sector QR0817 that it is 'Business'. At least BA's search is correctly finding a routing with part on J and part on F.

Still confused by your comment: "The HKG-DOH flight can be booked as first but is business with some first priviledges. The Qatar website confirms this from a dummy booking - bit strange in my opinion". I cannot replicate this on the QR website, and not sure where the "business with some first privileges" comment comes from.