I'll reiterate what I've said a few times recently - cashback via TopCashBack, Quidco etc. is never guaranteed and can be declined at any time, so should be viewed as a bonus rather than a reason to go with a particular retailer. I've had cashback come through on occasion, but I've also had it denied for really pedantic reasons.

Amex cashback, however, is a very different story as it is guaranteed with a qualifying transaction.

In your current situation, given the bonuses for booking through BA directly I would choose this option.