I can't imagine that Michael would ever have allowed his personal business to be aired in divorce court, and Kay surely knew that.

They must have made a private arrangement - something along the lines of Kay being granted a divorce in exchange for Michael getting primary custody (at least for a while - they did end up with Kay eventually).

Kay might not have really objected to that. As much as she wanted them out of the Mafia environment, there was always the chance the children would be targets for Michael's enemies. They would be safer on the compound.