Gifting/sharing app to Family child's iPad without App Sharing?
As you know, one of the benefits of Family Sharing is that one person (the Organiser) can take responsibility for purchases of all apps, books, music, etc. on family devices even when the users of those devices have their own Apple ID allowing contacts, messaging, etc. to be kept separate from those of the Organiser.
That is the method you should be using even if there are some frustrations.  You could just add your son's account to the 'family' leaving your partner to manage her own purchases (if that is appropriate for your family).
You can't just transfer/share an app from your Apple ID to another Apple ID. That would be open to abuse. Family sharing was introduced to satisfy your needs whilst allowing Apple to maintain control of the use of purchases.