I fail to see, from the multitude of venomous posts on this thread from a variety of people, where the problem is.
The only reason BA stipulate 'integrated' is because they can. I have flown with many pilots both modular and integrated and, to be honest, I don't feel the need to ask 'so, what form of training did you undertake?' at the brief and then expansively suck teeth if it was modular. There is NO difference once you have the green, foul smelling, plastic ATPL folder. Everyone is qualified to the same level, irrespective of how they got there the only difference being that the LHS has completed a command course with the company and the RHS has not and, possibly, needs to build experience.
BA use the integrated as a way to cut their apparent costs. That's up to the recruiters. If they want to stipulate 'only to have flown Cessna Citations with a purple stripe down the side and a leather interior' then good luck to them. They can. The 'stipulation' disappears as soon as you become DEP and then you can join the company on the DEP payscale.
Always remember, never recruit unlucky people by throwing the top half of the CV's in the bin!
Anyway, the whole point is moot as the entire airline industry is in a downward spiral and the chances of 'sponsorship' or 'cadets' being resurrected in the near future is about as slim as an apology from Gordon Brown for the state of the economy!
Good luck, chill out and welcome to the real world.