I know all those areas very well, having grown up there, left, moved back, left and moved back again....!!!! I have experienced them as a toddler, teen, 20's, 30's etc..
Given you are looking to buy a family sized house, I imagine you are in this for the long haul, and therefore I think you need to consider every stage of your family's life.
I would not recommend Brixton for the years when your child is under 18. It can get very busy and there is no space for them to roam independently or ride a bike etc. You always have to end up going to other areas. Brockwell Park is nice, but it can get busy with lots of randoms and some people wouldn't feel comfortable letting their young teens out there without parental presence. Your kids sound quite savvy, know how to use the tube/bus etc, but having a safe environment with space on your doorstep is priceless when kids are under 18. That being said, transport is great if that is a key factor.
West Dulwich is a slightly bizarre area. I would include the Village as part of West Dulwich (anything in the SE21 postcode). Transport can be excellent depending on where you are as North Dulwich and Tulse Hill station has very good rail links for someone commuting into Central London. Or, it may even be the case you are near the Gipsy Hill end, in that case, you have a 1 stop journey to Crystal Palace which has even better links than the former, and even better links than Brixton in my opinion. To get the real benefit of West Dulwich, I think you need to be walking distance to the Village or the College Ward area which has lot's of safe space.
As a young teen, I really enjoyed West Dulwich. It is safe and very green. The roads are quiet. I remember cycling on pavements close to Dulwich College, playing tennis at their grounds, generally just playing on the streets! There is a lot more 'space' in West Dulwich than the other two areas and I remember my parents not having to chauffer me about as much as friends who lived nearby in Streatham/Brixton, as they were happy to let me wander the genteel quiet spacious streets.
Herne Hill is a bit of a go between, not quite Dulwich, not quite Brixton. A good compromise if you like elements of each of those areas. It's not somewhere you go to visit, in the way that you would visit Brixton or Dulwich. It is quite an expensive area and I am not convinced it represents good value for money.