Lots of these moves were to reduce costs by creating a lower cost base subsidiary (such as Air Canada did with Rouge) where new pilot and cabin crew contracts were issued. This doesn't mean the prices reduced as well!

I don't think I have any first hand experience crediting to either BD or A3, but based on previous experience of others crediting to both those FFP's:

Austrian-Tyrolean - definitely no problem
Swiss-Helvetic - definitely no problem
SAS-Blue1 - Blue1 is wholly owned by SK, and is an affiliate member of *A. Of all the airlines, I'd be most confident of this one posting.
LH-Airbee - I'm not aware of Airbee (I thought they were Italian and went bankrupt!), but this sounds like a wet lease which should therefore credit as a LH flight.