Update from onboard BT using the (super duper fast) starlin wifi: Agent at KL lounge, after having some initial doubt, saw the marketed KL code and allowed me access. BT uses the Aspire lounge so I believe access on a BT marketed flight would not be possible.

On this topic, it seems that there is a difference between AF and KL, at least in terms of the pages: AF page explicitly lists lounge access when flying BT (based on this page) compared to KL which does not list it all (here)