How do I transfer my child's Apple ID to another Family Sharing group?

So, I am the family organizer for my kids' apple ID's. I need to move my son's apple ID from my family, to his mother's family. This is challenging as I just realized that, when I updated his email address, his apple ID didn't switch to the new address and the old address is a dead gmail account that no one has access to because it got hacked and frozen (that or my 12 year old is secretly a Russian internet troll mastermind).

After hours of unsuccessfully trying to switch his account using the email address I thought it was linked to, I realized when looking at my family section on my phone that his email was the old one. Brimming over with relief and enthusiasm, I thought I'd hop on my laptop, update his email, then all would be well. First I hit a roadblock where I couldn't change his email because stolen device protection was turned on and no one was getting notifications to authorize the change, after 20 minutes of internet sleuthing and 3 facetime calls we got that turned off but now when I go into his icloud account, I can just change the apple ID to the other email that's listed, I do have the option of changing the apple ID to a new email, when I enter the other email address that's listed, it prompts me for his password, which I enter, but then I get an error message that his email "can't be updated at this time". I can't switch him with his current apple ID as there is no access to the old email.

This may take the cake in frustrating technical problems, it's harder to update an email than it was to program a dungeon crawler in BASIC. Please help me, also, please tell me why this is so hard.

[Re-Titled by Moderator]