AIUI, it is not just a question of using the right GDS but also of booking the correct rate. The rate used by BA Holidays will in all likelihood be an ineligible IT rate. Pretty much anything booked through tour operator (and BA Hols would be tour operator in this context) will be ineligible even if the tour operator's products are sold via one of these GDSs.
That said, if you can convince the Hilton property to enter your membership number in the reservation, the odds are that elite perks will be granted even though the TandCs say differently (points, OTOH, will definitely not post).