Many thanks to CWS!
I'm just thinking of doing a commoner trip to the States at some point and thinking of booking HEL-LAX (once it hopefully returns in 2021) as BA by AY or try World Traveller with the Queen of the Skies from LHR, before they vanish (done First and CW, never WT or WTP - such a TP snob), as BA by BA. However I realised that BA coding excludes checked luggage with the cheapest tickets, even if it wouldn't be BA operated, but AY to LAX. But good to hear AA and IB at least would let me check in a bag with gifts/souvenirs on my trip as a commoner (well, a pleb with a status card), even when it's BA ticketed and marketed. OTOH, I highly doubt I will do it on AA, since I want to select my seat in advance and not as an extra per diem.
Too bad can't book IB by BA and use the OWE status. Or was this perhaps the exception: BA metal, coded something else and OWE from some other OW carrier - you get the luggage as per the status of your OWE card?
It's funny: done First, done G-EUNA as BA1 and done upper deck on the 747-400, but still haven't done any TATL as a pleb on any carrier. So it would be an educative experience and remind me that I must at all cost avoid doing it again.