I don't know of any apps that do this exactly, so maybe if you want to stay within YouTube you could try restricting to a playlist. But that still has the possibility of random videos playing at some point. I've also been looking for something that allows me to loop videos, although that's more for creating a TV-channel like loop for my grandparents.

To be safe in this situation I'd just use something like youtube-dl [1] and save filtered videos locally. Kids tend to watch the same videos over and over, so why not save bandwidth/battery life and store it locally when it's so cheap to do so. And since it's likely cartoons, the compression is better saving even more space. Tip: You can download playlists, so if you set a cronjob task to download a playlist you can add videos to it from anywhere you have access to YouTube.

[1] https://rg3.github.io/youtube-dl/