There is a transit area but arrivals/departures are not mixed. When I last flew there on BD it was about a 5 minute walk from the gate towards passport control - you take a right marked 'transfer' just before it and come out in to the main terminal just before security. There is a fast track, but it was closed when I was there on a Saturday afternoon. I waited a few minutes for security to clear and then went back through. I'd say it look 10-15mins total from landing to getting back airside. You could then sprint to the gate and maybe just make it with your PYOBP...BUT with no luggage checked, they may simply offload you and go.

Personally, I wouldn't risk it. Book the next BD flight and instead spend some time in the lounge on the Belgian Beer (even better if you're BA Gold too and can Open Doors your way in there ) and relax