Hi there. Would you consider living a bit outside the city? If you want REALLY good schools, I'd recommend living in the St Albans/Harpenden area (Hertfordshire, north of London). The trains from there go directly to City Thameslink station, which is on Fleet Street. Fast trains take about 40 minutes, and a travel card runs about 300 GBP a month.
The council taxes are higher because of the good schools, but my guess is that'd it be worth it if education is your main concern. There are a lot of young families in the area because of the schools. Also, living out here, it'd be fairly easy to find somewhere with a parking space, so I wouldn't worry about that. And there's plenty of green space for your dogs!
I think you could easily find something within your price range. Feel free to send me a message if you want any further info about the area - I've been living here almost a year (was previously in central London) and can easily say it's a very nice place to be.