Apple Watch App Store not working with Family Setup

Bought my son an Apple Watch in lieu of an iPhone. Set it up using Family Setup on my own iPhone. Some Screen Time controls are in place (we’ve been using it for several years with his iPad) as is 2FA. He is excited to download some apps on the Watch. When he tries to “get” an app, the App Store sends him to Settings to enter his iCloud password. We enter the password (tediously) using scribble, then return to the App Store and try to “get” the app again. It sends us back to Settings to enter the password again as if we had never entered it.

This behavior persists no matter how many times we try to download the app, making the App Store completely useless on my son’s watch. Similar behavior was noted here:

Can’t install apps on Family Setup Apple … - Apple Community

Seems like a bug or a design flaw. Yes, we know how the App Store is supposed to work on the Apple Watch so no need to post a tutorial on that. Would just like to see a fix or workaround for this problem — seems like a big issue if you can’t install new apps on the Watch.