Westminster take relatively few boys for sixth form. Roughly 10 out of the 70 strong intake. From observation they are quite a mix, some from overseas, some from state, sometimes one or two from Eton/Winchester, some from other private schools in London or outside. Of these 10, probably only 3-4 will be boarders.
And then within the group there will be a mix of humanities students and scientists, some are very very clever, others have real charisma, others are the sorts of kids who engage well in school life.
In short, other than those who are truly outstanding academically, it is possible but a long shot.
What helps is that the big intake of girls loosens up friendship groups so new boys seem to integrate well. Both my DC had a least a couple of good friends who started at this stage.
My DD loved her two years at Westminster, giving meaning to the phrase "having the time of your life". I am not sure that you need specific extra-curricular, though music, drama and debating are all excellent, and some of the sport can be very good. More important, I think, is a willingness to take advantage of all the school has to offer, whether it is Latin prayers, expeditions or whatever. The academic bar is high so there is no need simply to import "results". Instead they seem to like those who contribute inside and outside the classroom.