Definitely a weeping mulberry would be best. They create a wonderful hidden world beneath their branches, and could probably be climbed as much as possible for a dwarf tree. Just be sure you get a fruiting variety, I think there is one called 'teas'. Most weeping mulberry trees are fruitless (males), so you may have to get mail order from a good nursery.

Scheme basically not feasible as by the time the tree will be big enough to be climbed without being broken the kids will be many years older than they are now.

So you should probably get or build a jungle gym instead.

You could still plant a mulberry for fruit and see some production in a few years. But the tree will again have to have enough time to size up before there is any potential for a lot of fruit.

And mulberries are quite staining - you will not want to locate such a tree anywhere near features you don't want stained. Also when in fruit these draw numbers of birds that poop all over everything, including the tree itself.

Girardi is a dwarf mulberry. it is sweet with excellent taste. Some ten years old tree only 6 to 8 feet tall. Very easy to cover it with a net for bird protection.