On shooting varmints, this place is stocked with just about everything from air rifles to something big enough to drop a moose and can operate them all pretty well. I live on 10 acres, and I've given it some thought and have concluded that except for a shotgun, I could not safely shoot at a varmint running around on the ground. Between the homes, buildings, and livestock I can see and those I can't, plus places people, livestock and pets could be (and often are) that I can't see, there is no guarantee someone will not be in my line of fire. So I'd never consider shooting at a varmint on the ground with a rifle of any type. And again, I"m on 10 acres. Shotgun, yes (and have). Rifle, no. And that is assuming I"m around and could act fast enough to do something about an attack if I was.
I fully understand your concern about a 3 year old and an electric fence. I have a grandson the same age. That kids mother shares your concerns. That family also lost all their birds last year to a fox. The birds were free ranging in their fenced yard. The only reason it wasn't a coyote that got em was the fox got there first.
I too have a fenced in back yard, chain link style that goes to the ground and is about 4 feet high. But it has gates and in a few places, it is a few inches off the ground. I have seen cats, skunks coons and possums go through it like it isn't even there. A physical fence of that type may not even slow them down. Adjacent to that is a garden area surrounded by an electric fence. Within that garden area is a patch of sweet corn that to a raccoon is more addictive than crack. They haven't touched it, even though I know they are on the property each and every night. Trail cam photos prove it. So we know an electric fence works to keep them at bay. And compared to the physical fence, the electric fence isn't even a sieve. Unless it was hot, they could pass through it like it wasn't even there. But they don't. It isn't a physical barrier at all. It is a psychological one. One they won't touch.
So the trade off seems to be electric fence and the dangers it poses, vs. losing all your birds. That is why I say an electric fence is your solution. So find a way. Do put it outside your existing fence (there are stand off clips for both steel T posts and chain link fences that will put it on the outside). A wire can be run along the top high enough toddlers can't reach it. And it can be turned off during the times when they are outside. If they are outside, the noise they make may be enough to cause the varmints to keep their distance. But if that fence is left on most of the time, and a varmint ever touches it, that then becomes the mental barrier that keeps them at bay. As it stands now, you have nothing to hold them back. Nothing at all.