Left a few years ago but thought the post-T5 restriction on using BA lounges on personal staff travel was on F/J17 and greater (which covers non-management pilots) and if you are F/J16 (senior management, all animals etc) then lounge access is permitted, might be my fading memory though .
If one is a BA silver card or better holder then you are permitted to use BA lounges on staff travel regardless (lots of hotline commuters) or if you are accompanying a card holder as their guest.
edit: maybe SM grades and higher get Silver/Gold cards automatically so that’s how they get around the lounge restrictions