I'm completely befuddled. Why is entire built with ajax and why is the database search directly rather than use wp api? Shouldn't it be built with php up front, build in the children, just make them hidden, and only use ajax for modifying info?
When searching for children, it only searches post type pages, which means it doesn't cover custom post types.
if you query the posts with wordpress api, it would take care of that, and a whole bunch of other issues bound to arise.
get_posts($args) or query_posts($args) would do the trick.
Any chance you could post it on github so I can submit edits? Or should I put them here? Or do you have another place I could send them? Either way, I'm only going to fix that one issue, but it really needs to be fixed throughout the entire plugin.