I went to the International School in The Hague and had friends at both the British and American Schools. I'd say the British School is definitely one you'd have to feel comfortable with to attend, it's quite a homogeneous group. The facilities and staff are all great from what I remember but it is very Britain focused vs. the ISH and ASH which allowed more flexibility if you expected to move elsewhere at a later date. So the general rule was, Brits who knew they were returning to Britain and wanted the least amount of disruption went to the British School otherwise they'd attend the ISH which is also quite British in staff and structure.
I'd recommend requesting a visit at various schools ahead of time to get a feel for it and set out your needs. HSV as mentioned above is well known and popular with local politician kids and other intelligentsia but I'd warn that if your children either have learning disabilities or are of above average intelligence that school doesn't serve them well (friend had a terrible time with DS1 who is highly intelligent and sat bored for over a year before it being brought up, another has a DS struggling to keep up resulting in behavioural issues). The ASH has a special programme for children with learning disabilities that has a lot of parents flocking to them lately from all over the country.
Local schools are all of high quality and expat children are not an oddity but do keep in mind kids go to secondary school a year later than in the British system so I'm unable to comment how seamless that overlap would be for an 11yo two years down the line.
Living near schools is the general rule especially if you go to a local school, depending which school you go to Statenkwartier (the shops on the Frederik Hendriklaan that cater to expats being the main attraction), Archipel buurt (large houses and well situated within the city for work and schools), Duinoord (more affordable in comparison and well situated while practically Statenkwartier), Benoordenhout (especially for those working at Shell), Zeeheldenkwartier (less popular with expat families) and Wassenaar are popular expat areas but people live all over the place depending on budget. Wassenaar is a little bit more isolated (requiring more than one car possibly) and a more 'known' expensive area but it is very nice and green with the odd mansions here and there but very much less cosmopolitan. Those who can't afford the Wassenaar prices but want to be close to the ASH or BS tend to deviate to Voorschoten. Don't know if they still do it but the ASH had a bus service that catered to certain areas.
Here's a link for the municipality's info in English. In the left sidebar there's a special 'Education and childcare' subset for more detailed information about your options.