Book with BA and submit a claim against their Best Price Guarantee - see https://www.britishairways.com/en-gb.../price-promise

A bit of a convoluted route I regularly fly is ALWAYS cheaper on "Carlton Leisure" and every time I book I do it through BA at the higher price, submit my claim for the price difference and get credited the difference back to my card, usually within a couple of days. Its been anything from around £50 - £150 each time.

Obviously check its exactly the same flights, same times and booking class first and that it meets the criteria for the Best Price Guarantee.