I know this has been discussed on the Portland Chickens listserve but I've deleted the thread from my email. 
Lead can be passed on in eggs, and you can have the eggs tested. The chickens, children, and eggs involved were all fine as far as I remember. It was quite the hubbub on the list for a while though, as many (myself included) utilized repurposed/recycled building products in the coops/runs, or live in old houses where the chickens have access to siding painted before 1978 (when lead was outlawed in paint).
Here in Oregon I'd recommend you contact DEQ (Dept of Environmental Quality) as well as the pediatrician (non-emergently). There must be a similar program in NC; your pediatrician's office should know who the resource for the state is in NC. The good news is that the people on the listserve who had eggs test + for lead after contact with lead-based paint chips were able to remediate the problem by encapsulating the lead (painting the windows if the paint is in decent condition) or otherwise removing the lead in the chick's direct environment.  I like the idea of putting hardware cloth in the windows to prevent the ladies from getting to it.
Your kids will likely be just fine. Really. Lead is a pretty ubiquitous environmental toxin, and almost everyone has had some exposure. You don't want it, but there's no reason to panic. I don't take the attitude of "I chewed on my lead paint crib and I'm fine", but I do take the position that you do what you can reasonably do to prevent further ingestion. 
If your kids have very high lead levels (unlikely unless they're eating the lead paint directly), there are treatments that your pediatrician will recommend/refer. That's a RARE case that requires that level of intervention. If your children require testing, your pediatrician will guide you through the process. I had my toddler tested for lead after a dusty bout of remodeling in our home a decade or so ago. She was fine.
You know you've got lead paint and the chickens accessed it, but you don't know anything else yet. While you're in "let's-figure-this-out" mode, I would err on the side of safety and not have the kids eat the eggs for right now until you're sure that the problem has been addressed. I'd personally pay $200+ to have some eggs tested if I had documented exposure (but not the specific chickens) just for peace of mind. If they're +, you take care of the lead and then restest the eggs in a couple of months to make sure everything's OK. And yes, I'm an RN in my non-chickening time.
BTW, that is one breathtakingly beautiful coop. I see why the windows spoke to you!