I would recommend that you cycle to Bakken. It's a very nice trip and you can take the S-train home. Bikes are free.
The best time to visit Bakken is early evening, so start your bike ride from Copenhagen in the afternoon.
If it is hot, there is a lovely beach in Charlottenlund, perfect for children. There is a kiosk with the possibility of buying an ice cream.
When you arrive at Klampenborg, Bakken is to the left in the forest. On the right hand there is another beach, (Bellevue) so there is the possibility of another dip in the water.
On Bakken there are plenty of resturants, cheap than in Tivoli, but the quality is very different.