That's probably WT Plus. Never flown in that cabin. I see you are a BA admirer too. I salute you sir!
BA definitely does not have the cheapest J class. I have no idea when it comes to US airlines, but from the UK, a number of airlines provide cheaper premium fares:
Kuwait, Air India, PIA, Emirates, Gulf Air etc. In fact, my lecturer recently flew to NY with KU in Biz for a reasonable GBP807.
Best deals are found with Travel Agent's.
They have a few J class "bargains".