Hi ! You're really close actually, but you have too many unnecessary elements.
First of all, you don't need $refs except if you have to interact with the element by using vanilla JS (useful to bind some third party library events).
Second of all, you don't need the value property because you can manage what (and how) you want to send data to your PUT/PATCH route which updates the child status.
There are minor fixes that can be applied, but for that, you can check out the sandbox I created. Of course, you shouldn't need the changedChild data property. I only used it to output what's happening visually. Also, I changed you href for # because it's just a test environnement. I also mocked some children because I don't have access to your API.
Basically, what you need to do is pass your current child as a parameter to the function you call on change. In my sandbox, I renamed the function to updateChildStatus, because that's probably what you're gonna do in that function. That being said, that's the syntax:
@change="updateChildStatus(item)"
Then you can accept the item as a parameter in your updateChildStatus method declaration. From there, manage the information to transform it as you wish and then PATCH (or PUT) it.