[Fire Q&A, Help & Troubleshooting] ArashiPrime:
Actually just registered finally so I could reply to this. I wrestled with this a bunch over Christmas - the key to doing this is an app in the Amazon Kindle App Store called GoToApp (I used the free demo version). Since it is in said store, you can assign it to your child account via the Freetime app on the parent side. I also also temporarily added a file manager to the child account as well - I used ES File Manager, which is also free and in Amazon's App Store, though I believe any file manager would work. Since they are now assigned to your child account, they should show up on the carousel in FreeTime mode like any other app . I then loaded the 4 APK files needed for Google Play and the YouTube kids APK onto an SD card. You then go into the child account and install the apps, using the file manager to get into the SD card. Once done, go into GoToApp - this app is essentially a launcher, and you can add YouTube Kids into it and let them launch the YouTube Kids App from within GoToApp. Both my older and younger sons very quickly learned that essentially GoToApp's icon = YouTube now.

A few other notes on this, however- if your kid is smart, they can use GoToApp for other things. I had manually installed a file manager and he figured out how to unhide it in GoToApp and then went to town. For my 3-year-old this was obviously a non-issue, and it has worked fine for him.

Also, the YouTube Kids app is very, very finicky when trying to get it installed. I found a specific version that worked when nothing else did. I don't know that version off hand, but can dig it up later if needed. I essentially just kept going back versions via APKMirror until I found one that worked (though once installed I never have had to update it, and it still works fine). I also found that installing the Play framework seemed to work best if I installed it only on the kid account and not the parent account - one tablet I had installed it on my account first, and then probably wiped it a dozen times overall before it worked decently. Only installing it on the parent account would give errors when it was then run via the kid account, and I had problems when I installed it on both. Once I figured out the way to do it, the other tablet was fine; I didn't have any problem with that tablet.

In the end, though, I ended up reverting back to a normal kid account for the older child - the combination of the backdoor combined with a lessened amount of parental controls led to him getting a little too into it some nights. I think that both of those issues are really more my fault more than anything else, especially the latter. What we did end up doing was putting YouTube Kids in the parent account and locking down as much as we could (can't open the Play Store or Settings without a PIN and so forth), and he now has to ask to watch (instead of being let loose with it) and is at least semi-supervised now. This also let me take better advantage of the FreeTime controls. On the other hand, the younger one still has this GoToApp setup 9 months later, and it has worked well for him with no problems to speak of.

Anyway, novel of a post, but if rather have someone learn from the mistakes I made and not have to put hours into it like I did.

EDIT: For some reason I thought that the above comment was second-to-last without any reply to it. Oh well, hopefully this helps someone.

Might try Google Play Services 10.2.98 (230-146496160) instead - this is, I believe, the version I installed on my kids' tablets (one 2015 Kindle Fire 7" and one late 2016 Kindle Fire HD 8", if memory serves). I actually just did a wipe/reinstall on the HD tablet last week and the same file worked for me on the parent side at least (didn't reinstall on the kid account though).