Ive found something that works, hopefully it helps you as well. This was after hours digging and messing around with settings and having to input my password countless times through the terrible scribble pad on the watch.
First of all- The Apple watch is set up through family sharing as a stand-alone watch without an iphone "companion". It appears this brings up limitations for that apple watch.
Option 1:
Idk how this app actually installed, but it did. It wasnt immediately either, rather it took a day or two for it to actually appear on the watch.
-- I searched for the app on the app store, clicked on "share", sent to apple watch se (with cellular) through a text message. My son clicked on it, and i input the password about 6 times where it just kept returning to the "get app" screen.
I gave up and continued another day. Appp hadnt downloaded. A few days later, my son tellls me his app doesnt work, it was "watchtube" in this case. It had finally downloaded but was faulty. Never figured out the problem.
Option 2: Best option.
I believe to have changed some settings allowing purchases in trying to avoid inputting the password so much this time. Read up on family sharing and had an idea.
-- first download the app on your watch ( its easier to actually use your iphone)
--Make sure it shows up on your family share, previous purchases, etc
--make sure its compatible by checking that it downloads to your watch first.
-- Using the kids apple watch, go to the app store, scroll down and select "family purchases", then select "your name", under which your apps shall appear.
-- click the app and select "get"
-- follow proompts to enter the password for the kids account.
-- set the watch to charge.
-- review the watch for installation.
Good luck