I'm a fan of Kamakura. Some people here disagree with me. I find it to be a very classy and interesting place but it's proably less kid friendly than Hakone since it's a lot of temples and walking... The kids might like going inside the big buddha and washing some money at the zeni-arai-benten temple.

When it comes to beaches ... don't get your hopes too high. While Kamakura is by the sea, with easy access to beaches at Yugihama, Zushi, Enoshima etc. you will find the beaches to be crowded and dirty. There are nicer beaches if you go all the way down the Izu peninsula, but you're still going to be in the middle of a big crowd. I think it will be especially disappointing for someone coming from Bali. Perhaps you should consider a mountain trip?