One of my closest friends here, also an American, shares the identical predicament as you do. It's now been over three years since she last saw her kids back in the US. And just like your ex her ex also doesn't allow the kids to come to Qatar even for a visit. I know how sad this all makes her, though she does have kids here from her husband, all her kids from her former marriage are back there. The fact that she is left out from so many occasions and milestones in their lives is something she can never truly ever get used to. She only bares with all patiently.
Truly you have my sympathy and well wishes. As a mom I cannot even begin to feel the pain you brave ladies must feel,as I know how much I miss my daughter if she's even away from me for a day. Insha Allah I truly hope things will work out such that you will get to see them sooner. God is truly able to do all things! Take care.