We recently went through something similar. I find it a bit odd that the missionaries are not instructed in the baptismal interview, to explain exactly what information the Church will keep in its records (the baptismal record form in the missionaries area planner app. doesn't list everything the clerk needs), to resolve any concerns prior to baptism. That said, when a convert is baptized, the Bishop and Clerk do receive an email requesting they complete the membership record, which includes "Please review the pending record to verify the accuracy of the new member’s personal information, including name, address, baptism and confirmation dates, and family relationships." So family relationships are intended to be included in the member's record, which in practice means children's names and birthdates.
Perhaps the Bishop could meet with the new member, listen to the concerns and questions, and possibly offer to also meet with the mother of the children to allay concerns about what information is being retained, who has access to that information, and what information will never be stored. There may be more than simple privacy concerns, and if for instance the mother has strong feelings about her children not participating in the Church that the father has joined, it would be good for the Bishop to get acquainted with those details.
As a stop-gap measure, I believe (?!?) the clerk should be able to go into the record approval, un-toggle the "has children" section (or whatever it is), and complete the record without any listing of children. Then handle that aspect as it gets resolved, but at least the record would be created. But that's just a guess as to whether that's possible once the "has children" toggle has been checked.