FYI, you can edit your original post to bypass the character limit. No need to self-replying.

On topic, based on This GameFAQs board, the game is registered to your account at the time of purchase, so if you bought it on your PSN then they need to be logged into your ID to play. Based on other topics of similar inquiry, neither SE or Sony will be particularly helpful in getting the purchase transferred. You might be able to get a refund through Sony, though, if you can convince them it was an erroneous purchase.

However, should you have no issue with your child playing on your account AND you haven't linked that purchase to your personal MogStation account, then they can still play with you. But based on what you've said it sounds like you entered your MogStation account information on your account already. If you can delete the service account and unregister it, I have no idea how.

To answer your other question on whether the game is cross-platform (it is). My wife plays on PS4 (transferred from PS3 during the free upgrade) and I'm on PC. We're on 2 different accounts, as trying to log in with the same SquareEnix ID will boot whoever is currently online, if I'm not mistaken.