Hi,
My Cambodian wife (we are married in France) has a daughter with a Cambodian man (never married). The daughter is 10 years old and we are moving her here to us in France(she lives as many other kids with grandma, but sjes getting old and we need her to get better education than in her home).

The father has never been present, he is not in the family book, the kid has the mothers last name, she has a passport. The family book lists the mother as head of family.
The father has put his name on the birth certificate. But nowhere else.
Now the french migration agency says they need a paper that proofs that the mother has sole custody of the child.

As I understand it, the Family Record Book setting her as head of family is exactly that.
Or do we need to go to a lawyer and get a court order for this?
We have already waited 9 months for the French Migration agency and now they dump this on us, last minute.

Any advice on what to show them as proof of the mother having custody over the child?

I suspect you will need a stamped divorce settlement/court order as evidence for any foreign agency. I have this for my son, it is clearly defined in the ministry stamped divorce document that can be used, etc.

I'm sure with all the background you have given that this is fairly easily obtained from the Municipal Court in PP. I think your wife can probably find a legal/cheap route to obtain this type of court order without a lawyer, but if you need one, I used CSP & Associate who were superb & spoke clear English. https://www.cspalawfirm.com/