If (and its a big If) BA were to purchase a higher stake, I really doubt it'd be the death of BE.
Whilst everyone likes to bring up the whole Brymon, Manx & Cityflyer thing, it really isn't a fair comparison. Those airlines were tiny when compared to the likes of BE, and they were facing competition from, well, BE (and its predecessors) and a myriad of other airlines. BE faces relatively little competition on the routes it operates compared to the bad old days.
I think there is perhaps room for BA to purchase a higher stake, and integrate more. But whether it's the right decision at this stage is open for debate.