The amount of connecting traffic on Aer Lingus' DUB-LHR route is completely overstated on this and other forums. The vast, vast majority of traffic on the route is point-to-point. So should BA decide to operate the route using their own metal, just offering long-haul connections through LHR isn't enough, they'll have to compete for O&D traffic as well, which they won't be able to as Aer Lingus has a much lower cost base than BA and as such can offer lower fares, and arguably has a better suited product for the route.
As for the BFS-LHR route, this would have a much higher proportion of connecting traffic so obviously the code share is more important, but you'll find a lot of carriers are now adopting a 'metal neutral' policy, whereby they focus more on offering the best timed flight for the customer than feeding passengers into one particular hub or onto one particular airline. So under a 'metal neutral' policy, BA isn't going to force all its passengers to travel from BHD because it owns BMI, it'll also offer flights on Aer Lingus from BFS if that offers a better connection (assuming BA maintains bilateral relations with Aer Lingus).
So this idea that because BA have bought BMI, they're going to abandon their code share with Aer Lingus and force everything through BHD is nonsense. BA would effectively be abandoning the whole island of Ireland, because Aer Lingus would inevitably retaliate by abandoning the code share on its other LHR routes and transfer frequency to other hubs such as FRA, AMS, and CDG which would be bad for BA, who didn't just spend €210 million buying BMI only to use dozens of LHR slots on routes to Ireland using its own aircraft that it doesn't have...