I ended up taking a risk and changed the email address on the AppleID I was having issues with and then created a new AppleID using the work email address.
That has resolved matters.
I have then been able to log in to the App Store using my personal account, so I’ve got the best of both worlds (no iCloud sync for apps, as it’s different iCloud, but that’s not the end of the world and allows me to keep work and personal data separate).