I know you specified Aman versus Four Seasons, but with three children, I would, instead, opt for The Legian (GHM - little sister hotels to Aman - http://www.ghmhotels.com/en/legian-bali/ ) for your beach experience and then The Chedi Club Tanah Gajah in Ubud - http://www.ghmhotels.com/en/tanah-gajah/

Both resorts are superb. The Legian is a seriously good beach resort which caters very well to children (Kid's Club, pools, right on the beach) but is also superb for adults, with excellent food, spa etc. The accommodation is very generous in size and with unparalleled views of the beach.

The Chedi Club Tanah Gajah gives you a genuine experience of rural Ubud and has some of the best food on the island and extremely caring staff, so children are really well looked after.

I have stayed at all the high-end resorts in Bali, many times, and these two would be my choice for you and your family.