Booking it on the phone is the big clue here, since an agent can combine EI prime and BA, whereas ba.com tends to favour codeshares. The big downside to that would be that you won't get any TPs for that leg, and few - if any - Avios. But for a redemption they have to go into prime, which I should have noticed previously, you can't codeshare on redemptions. And no TPs or Avios in redemptions obviously. But yes to lounge access as a BA Gold or Silver, and yes to the checked bag.