Bromley mum here.
If you like the Palace Estate lovely but a bit pricey, have you looked at Chatterton village? That's where I live.
Whitehall Park is lovely - there isn't such a great park in the Palace Estate. And you're right next to Norman Park too.
We're in walking distance to the station and in the catchment for Raglan, Bullers Wood, Ravens Wood, Bishop Justus and Ravensbourne.
I do second what other mums have said though, be very careful with the catchments because they are tiny. That's the biggest difficulty you face finding a house around the area.
The houses around Chatterton Village tend to be slightly smaller than in the Palace Estate but therefore they are also slightly cheaper.
Chatterton Village itself doesn't have any exciting shop but you do have all your essentials including a Co-op, a corner shop, a DIY shop, a salon, fish and chips shop, an amazing indian takeaway as well as a really good indian restaurant, a very nice family run cafe... The pub in the village isn't the best but check the one in Masons Hill, soon before Homesdale Road (I'm awful with names, I'm afraid). Oh and that one is opposite a little shop where you can find very peculiar beers.
If you'd rather stay closer to Bromley South station (Raglan is under a mile to Bromley South and if you don't fancy walking there are about 5 buses heading that way), then you could be in the catchment for St Mark's which is a really good CoE school.
As you also named catholic schools, the best is St James' which is in Petts Wood - chatterton is allocated 2 spaces per year in St James though you'd have to commute to take the children there. I live really near to St Swithun's catholic church.
There is also Virgin Active two roads from St Swithun's if you're into that.
Two allotments: one by Virgin Active (oversubcribed) one a bit further down in Turpington lane with spaces.
If you go to the Trinity Village (the new development) you may end up in a not so nice school. And you'll need to get the bus to go to the station.
I'll be happy to help so do text me if you want some more information. I hope you find this helpful though.