Where exactly are you moving to? CP and Sydenham is a HUGE area.
Moved out of the area a while back, but St Barts is always popular, similar area to Eliot Bank and Kelvin Grove. Lots of people I know love it. Tiny catchment these days, though, and depends on sibling numbers!
Eliot Bank is relatively popular and has a decent reputation. I know parents who like it there and their kids are doing well.
I don't know anyone who put Kelvin Grove as first choice. It's a large school with a lot of SEND, lot of children with English as an additional language,and diabolical parking. Most of the parents locally that I know opted for EB or St Barts instead.
In FH, I have a friend whose kid is at Horniman and very happy, but again it is a tiny catchment! Fairlawn (also FH) has an ok reputation (and no uniform!). You would be too far from Kilmorie or Dalmain to get into those two, which are both popular.
Mixed reviews on Adamsrill from people I know as well- some love it, some not so. Can't comment on the catholic one on Sydenham High St, St Michael's or Haseltine (all Sydenham)
Depending on where you live Dulwich Wood may not be one you can get in to if you're more towards Sydenham/Kirkdale.
It's worth joining the local mums/parents groups on Facebook as they are really helpful with info (look up Sydenham/Forest Hill/SE26/SE23/ East Dulwich mums/parents).
Crystal Palace I don't know much about, but I know a few people with kids at Paxton that are Ok with it.