Schools are a very emotive topic because most people will think that the one where their child is at to be the best, so I apologise in advance if I offend anyone.
Of course if you don't mind paying private fees, please do send your child to private school, because there is no doubt that it will buy you better facilities and higher teacher to pupil ratio. However, if you can get into Belleville, please think of what a wonderful education you will get for free. My daughter is now in Year 2. We have had a few kids each year transfer in from Dolphin from year 1 (because that's how long it took them to get in) and the parents felt Belleville was as good, or better if you consider the facilities, outside space, and the additional clubs / activities that are available (of course all is on hold at the moment but hopefully will restart soon). It's an academically strong school, and whilst the class size is big (30), your child will develop different skills than in a private one (e.g. having to find themselves and develop friendships in a bigger group).
Whilst a lot of the parents (and pupils) live in terraced houses in catchment, i.e. are upper-middle-class, there is certainly greater diversity than in private schools. I also find the parents very down to earth; I am sorry to say, but I found e.g. when the kids were at private nursery (they went to two different ones), that some of the other parents were quite snobbish / standoffish, which I never found at Belleville.