I have no idea if any standard dowel would fit the id of a #4, but here's something I did on #7s or 8s years ago: When Bear first started selling glass arrows there were no glue in inserts-they would glue a short piece of dowel into each end of the shaft and sand/grind a taper that included both glass and wood. I still have a couple in the collection. I did the same thing on a homemade taper grinder similar to a woodchuck and they worked just fine. Thing is the dowel probably doesn't need to fit real tight....you'd be truing it as an assembly when you grind it. Especially for a kid's arrow, it's likely to be fine.
If you doubt it, or if I'm not explaining it well enough, I might be able to post a picture.