Some of these things get booked up quickly and in advance, but there are quite a few more than I will list below, which are more HOP based:
Scout Hut, St Saviours Church, Brockley rise - always very popular and I think one of the cheapest to hire.
St Saviours Church Hall
Hope Centre (Ewart Road I think)
St Hilda’s Church Hall (more Crofton Park)
Depends what you are looking for, but I’ve been to all the above (bar St Hilda’s which my other half has been to) and all have hosted successful birthday parties for that age.
You can also hire areas in Private Schools, and in Gyms (I went to one in Deptford Leisure centre).
We had our daughter’s 1st birthday party at Holy Trinity Church hall, se20 (not se23/26 but just down the road - we live in Sydenham). It was great - £25ph and we took our own food/ drink/ decorations (they have nice bunting up at the moment).
Forest Hill Methodist Church on Normanton Street (really big hall, but also smaller rooms)
Sydenham Community Garden (medium size but a lovely garden)
Seniors Centre on Stanstead Road
Mayow Park - Grow Mayow Community Garden (better in summer as not much indoor space)
Holy Trinity would be my favourite. The only thing I would say is that it is a bit hidden but you can walk down Sydenham Park off Kirkdale. The second one is the community centre at Jews Walk which is on the other side of Kirkdale.
The last one is the community centre in Lammas Green which you can access via Otto Close off Kirkdale.
I have been to parties at all venues but my preference would be Holy Trinity in the winter and Lammas Green in the summer as the kids can run around the green afterwards.