There is another option open to BA, which is only to sell a certain number of hand-baggage-only fares on each flight on the likelihood that those customers are far more likely to take up space in overhead lockers than those with a baggage allowance included in their fare. I doubt they will, as they may see this as making them uncompetitive on price with easyJet & others. A commercial problem therefore becomes an operational problem.
After the year they've had, with the bashing over the M+S BOB product, the IT failure and the whole hue and cry about the brand being devalued, I am surprised someone has taken the decision to press ahead with this. It is a risk that I can't see (as a customer) they need to take right at this time. Get another thing like this wrong with adverse publicity and it could be the straw that breaks the camel's, or rather CEO's, back.
Come to think of it, I suspect there may be several thousand people at LHR and LGW who will go out of their way to ensure it doesn't work in the fervent hope that such an outcome would prevail swiftly afterwards. AC can't afford another screw-up on his watch, and I would guess many of his employees think the same, although viewing it from entirely opposing angles.