Fast food ? Virtually never. Maybe a handful of times a year tops when convenience wins out but that's definitely not a "once every..." thing, it's a "we're stuck in the airport and there's nothing else" scenario. Well, we did have fast food last week because my brother was visiting (extremely rare occurrence) and he fancied some but that was the first time in at least six months, possibly more. Though to be fair they're very young still so this may change in the future I suppose. Over my dead body.
You need to be firm and say no. If they won't eat what you serve them, there's nothing else. That's the rule in my house. You don't control what the kids eat but you do control what you offer them. One skipped meal won't hurt them, I promise. Fastfood every day, on the other hand, might.
If school asks (though it's none of their business) why you didn't feed them, say that you did feed them, they refused to eat. What were you supposed to do, stuff it down their throats ? If exH complains takes this opportunity to remind him that regular fastfood is unhealthy and doesn't replace regular parenting.
If the kids make a scene, so what ? Let them. Because right now you're teaching them that with enough whining and making a scene they can get whatever they want and they're in charge, not you.
You need to break this dynamic now.