Is a mother entitled to custody of her children after marrying someone other than their father, taking into account that she lives in a Western country that legally grants her this right, and is it permissible for her to follow these laws if she believes she will provide her children with a better Islamic upbringing?

The mother has the right to custody of the children unless she remarries. If she remarries, the right transfers to her mother. If [the maternal grandmother] relinquishes [custody], the right transfers to the father. The mother is not permitted to infringe upon the father's right to custody unless he is a transgressor (fasiq), in which case she may then utilize the law to retain the children.