I have seen a school advertised called Window to the World, located in Kalibukbuk (Lovina). According to their FB page, they are a non-profit organisation and are supported by the Rotary Club of Lovina.
I know a few people who have sent their kids to that school and have only had positive things to say. When I first moved here there were some people in Rotary club who I preferred not to meet so I didn't get involved in that.
There is a Motessori school in Singaraja
https://www.montessori.one/
And bilingual kindergarten:
Aura Sukma Insani Bilingual Kindergarten
There are some good after hours schools, like Kumon, reading, English and math classes. There is a good swimming club if your kids are into that also.