I think some people overstate the importance or relevance of BA's share in Flybe. It is small minority stake with no board representation. It was acquired as "payment" for BA Connect. It's not like BA went out and bought shares because they had an interest in working with BE.
I doubt very much whether BA and BE work closely on anything and I'd be hugely surprises if BA were to do a deal and reduce capacity to allow BE to move in on what is supposedly BACF most important route.
I could of course be completely wrong... Aviation is a funny business and odd little deals have been done in the past so who knows.