I can't see how it could. When booked together the booking is made through BA Holidays which is effectively a travel agent. When booked separately the booking is made with BA directly, and adding a car later can't change the booking from direct to TA.

For what it's worth, I book most of my trips on BA.com and can count the number of POUGs I've ever seen on long haul trips on the fingers of one hand. (And I'm talking about simple ex-LHR r/t bookings with no obvious disqualifier for POUG eligibility).