Unfortunately, you can't do it that way. Only apps from the Amazon App store can be shared across accounts on the tablet. Because they are linked through the Amazon App Store when "shared".
However, over at the XDA Developers Forum, they found a way to install Google Play Services and the Play Store directly on the Child Account on the Fire tablet. But I don't know the process. I've only had one of our tablets with a child account, and I looked into it then, but gave up - that was several years ago and he aged out of the child side anyway.
To look into it further, I suggest checking out the XDA Developers Forum, searching for the thread about the "Fire Toolbox" in the "Fire HD 8 And HD 10 Android Development" forum section. Search that thread for child account, and there may be something about it there, or a link to another thread about that specific situation.