Bet Yosef has nothing to do with the shechita. It's a higher level of glatt.
A shechita crew is responsible for the meat from live animal until packing, including checking (Inc lungs) and salting.
In any case, I think OP was referring to the hechsher not the shita. I thought u were saying there is no shechitaa done by BBY.
I doubt RYM exclude BBY products, I never checked. Kashrus while eating out vs home is a topic that I dont think has been covered here yet, IMO it's very OK to have different standards for the two.