I recently had a lovely time at Mothercare in Discovery Mall, Kuta. Also found lovely things in Matahari Department store, again, in Kuta. There are a number of children's clothing shops in Bali which I thought were a bit pricey for a baby as the one thing babies do is grow! Also in Centro there is a store "Bali Zen" where I found some bits and pieces. Have fun.
You could google 'children's clothing in Bali' for other ideas.Edited: 03 August 2018, 07:23
Kuta Kidz used to be good, but I they either closed down or moved away from Bemo Corner in Kuta. Indigo Kids is still in Kuta. There are also babies and young kids clothes shops in Legian/Seminyak on the main road but I don't recall the names of the shops.
Brilliant info, thanks SurfingWild.