The chances of getting into BA from an integrated school (OAA as my example) is slim especially at the moment as they only take about 10% of the graduates from OAA a year. They source the rest of their pilots from the other integrated courses and of course DEP.
I wouldn't base your training choice on what you have heard xyz airlines may prefer. Try to base it on your own circumstances both financial and what suits you. Can you take 18months off work, 60k loan, spend 4/5 months in america or are you more suited to distance learning, fly when you can afford to and keep the debt to a minimum.
I chose integrated because it suited my learning style, I work best when im pushing myself with time constraints and goals.
Nick