It is complicated and that is why I appreciate the help.
Here is the way that I see it. I would be able to use my HSA for my child even though he/she is not a tax dependent. HSA's specifically address the issue of children of divorce and that my child qualifies as if her mother and I were not divorced.
I keep all the records of the child's medical expenses just in case I am questioned by the IRS.
The double dipping would be happening by my ex. The FSA is in my ex's husbands name and provided by his employer. My ex does not work. That does bring up an interesting point, I don't know about the issue of whether our child is technically a dependent of his. They do claim the child on their taxes and file as married.
This is important to work out now because I don't want to get into any trouble with the IRS because of something my ex is doing and we seem to be having more and more medical expenses come up for the child.
edit: Yes to answer your question Leevena, I am reimbursing my ex with my HSA after she provides me with the bill after insurance is applied and shows me her payment receipt. I have noticed in the past that some of the payment receipts say, "FSA" on them and that is how I know she is using her husbands FSA to pay.