I would suggest ED, particularly the top ends of Upland and Friern Road, where the two roads meet. It's a no through road so basically makes it semi-private with no through traffic other than cyclists.
The gardens are larger than your typical ED gardens, you are 1 minute to Peckham Rye park which is a great outside space with 2 great playgrounds and a decent coffee shop, plus the frequent busses up/down Rye Lane (12,197, 63 and 363) can get you to Peckham station in less than 10mins. On top of that you are only 5-7mins walk from Lordship Lane and all that has to offer, plus can easily get to Peckham for its fantastic restaurants (the best in South London) and bars.
You can get to the Dulwich schools driving or on the P13 bus at the end of the road. Or if it's nice weather, walk/bus up Barry Road and then a quick stroll through Dulwich Park.
You probably wouldn't need to pay your top budget either! But could do if you got one of the villas on Rye Lane itself.
