I’ve had family and friends in both schools. Studied in Beaconhouse myself, so there may be an underlying bias to what I say but I promise I’ll try to be as objective as possible.
From what I’ve seen when I compare both schools, Beaconhouse comes out on top, however this was only due to recent changes in strategies and methodology. I remember both schools being equally good while I was in elementary and middle school, however Beaconhouse’s reformist and progressive policies during my high school years helped it outshine way past City. This is an absolute objective opinion, you can notice the difference yourself when you compare students from both schools. Beaconhouse adopted a more well-rounded approach while City focused on more exculsive and conservative approaches. The lack of co-ed schools established was alarming. Beaconhouse has no male/female only schools (to my knowledge); I remember it laying foundations for new clubs for debate, literature, arts and craft, etc. (something unprecedented in the study-only schooling environment of Pakistan); It has been a champion in foreign (and local)-exchange programs; and it has excellent teachers and coordinators (most trained abroad) to supplement a great educational experience. It also charged lesser fees for more services while I studied there (I compared with my cousins studying in City at the time). My cousins were pretty pissed when City introduced new uniforms (mandatory* uniforms) for various physical activities that could be done in casual trousers and t-shirts (just like in Beaconhouse). During my high school years at Beaconhouse, I noticed the simple yet effective policies that, when compared to City’s exclusive/exquisite but redundant policies, surpassed City by a huge margin. There was no rivalry in Beaconhouse when I was there, not to sound pretentious or anything but we had nothing really to compare anymore. City wasn’t just following up well and had diminished itself from being ranked as a worthy competitor. My older cousins that studied in Beaconhouse back in the day had a rivalry with City, not us, Beaconhouse was already ahead of City while I studied there.
Oh and not saying that City can’t make a comeback, it can (with the ridiculous money they charge), but for now; Beaconhouse wins.