I have a Fuji Ace 24 with about 36cm bars, but I tried hunting for similar bars earlier, and they are hard to source.
Have you seen this Gitane for sale with 20" wheels?
Gitane Midget/Junior Racer
I suppose one option would be to start sourcing bikes that you could rebuild over time.
So, hunt for a good 24" or 650c bike now, and move the bars over to the smaller bike... then move the bars back as the kid grows.