Ok - that's a lot clearer [plus I have finished my coffee now] 
In your shoes, with three children who appear to be exceptionally lucky with their schools I would be inclined to take the long view on fees:
Assuming there is an actual way to get all three DC to school if you take the job, then I would take the new job [and get the train]. It would be worth establishing now when they offer the job to you whether or not you can work from home one day a week or do a flexible hours arrangement 8-4.30 for example. You won't see your kids in the morning but you could be home in time for dinner.
DC3 is currently at local 'Outstanding' State Primary School with a nice normal commute in the same school their siblings went to. It's presumably good if DC1 got into a great indy school from it and you don't say that DC3 needs any additional support so I am assuming not.
DC2 is currently at local 'Outstanding' State Middle School (soon to transfer, but where to?). Apply for the same scholarships as DC1, but the school may well give a bursary or partial scholarship because they are siblings. Use your additional income to top up fees if required.
DC1 is at our County's top achieving Independent Senior School (with hefty scholarships I might add).
Your DC3 is in a v good school and is happy there? A higher income gives you and your DH more flexibility with schooling options for DC2 and DC3 in time.
I would only consider it if you were going to be taking DC2 and DC3 into the City every day or there is a real issue with wraparound care for DC3 if you take the job. An au pair who can drive your car or after school babysitter/nanny would give you a lot more flexibility though.
Middle school is a new one on me - is that 11-14 or does it cover GCSE's too. By Senior school, is your DC1 heading towards A-levels? If so, it won't be long before an additional car and insurance becomes necessary/desirable.
Just my view - def take the job though :) It's a beautiful City.