@Elephant - congrats. Hope your son enjoys it. We will be down to a single boy there from the summer.
No idea about Catford to DPL except for what Citymapper tells me but... our experiences of travel follow.
Bus - 202 drops off at the top of Dulwich Wood which is a very steep hill up from Sydenham Hill station.
Driving - The route from Catford probably won't be that bad given my rare experiences driving to school. The issue will be parking. There are plenty of people dropping off in the morning so Bowen drive gets packed out. Alleyn Park is a parking lot for Sydenham Hill station as well I think so don't expect to park close or easily.
Sydenham Hill - very close to the school - just a short walk down the hill (steps out of station are no fun though!). Roughly 5-7 mins - you have to walk all the way around since they closed the gate on Bowen Drive. Probably the closest public transport.
Sydenham - Overground station - 450 bus drops off near to Sydenham Hill station but bus is very erratic in schedule and you still have a walk to do. You can't rely on it imho. We take a taxi from the firm at the station three days a week when the boys are not boarding, we book 20 mins in advance and they have rarely let us down.
Gipsy Hill - a long walk but doable with a scooter, except for uphill going home.
For me half the fun is the drop off over an unhealthy breakfast and coffee so perhaps get there early and have a moment to sit together?