Hmm, I have been camping a fair bit in this area but I must admit I haven't been to a site that ticks all your boxes - I am sure there are some though. Here are the ones I have visited recently:
Forgewood (Eridge, Kent) - £10pn adults/£5pn children 4 and over. Good facilities in the top field, no play area but children love the woods and the open space. Not sure about walkable pubs but the site has a restaurant.
St Ives Farm (Hartfield, Kent/Surrey) - £9pn adults/£5pn children 3 and over. Couple of lakes onsite so you need to keep an eye on the young children. No play area but, again, our children loved all the space and going to look at the ducks. It is walkable to pubs and there is a nice playground in Hartfield opposite a good pub with a garden. Facilities are not the best but there are showers and toilets - some are just portaloos.
Chafford Park Campsite (Ashurst, Kent) - £10pn adults/£5pn children 3 and over. No playground but there is a wildly popular rope swing and lots of long grass to run around in. Nice facilities although a bit limited with only two loos and showers for each sex. I gather it is walkable to a pub but we didn't try. Lovely owner who is relatively new to the whole campsite business so happy to listen to suggestions.
Manor Court Farm (Ashurst, Kent/Surrey) - £10pn adults/£5pn children 4 and over - no playground but rope swing and farm animals. Nice facilities - not sure about pub.
Park Farm (Bodiam, East Sussex) - £8pn adults/£4pn children 4 and over - play area and decent facilities. Also a river with a rope swing and a nice walk along the river to a pub and Bodiam Castle. This site used to have a bad reputation for noise and trouble but the people who run it now have clamped down a lot on this and it has improved greatly. That said, it is a big site and there are a lot of people camping on it so it will always be noisy to some extent. It even has EHU now for those who want it.
All the above sites also allow campfires and all will sell you wood for that purpose.