In my web app, I just say "X more replies..." on that button using the child_count property. It really doesn't matter much to the user how many comments are going to be loaded next in my opinion, and that keeps it simple on our end.
If you really need to know this, you could consider faking it a bit by fetching an extra level/depth each time the user presses that button. Then, just hide that extra level until the user presses the load more button, and manually count how many comments there are in that next level. This way you can have the correct direct descendant count, each level will load instantly to the user, and you'll be fetching the next level in the background.