The Qatar A320s sent to BA don't have the lie-flat seats. They are the older models.
A7-LAA was Al-Maha which is now in Qatar colours. Again, no lie-flats. Rumours are Qatar wanted to send all the Al-Maha aircraft but BA objected as they thought it would confuse passengers. Hence all the aircraft sent are in full Qatar grey livery.
There are various pictures online including on the BA Flyertalk site.
Champ