I think it's sorted now, although not entirely sure why this worked this time when it didn't work earlier today.
I went into the company via QBOA in the QBO app (rather than web site) and it let me add another user to the client. Not sure how, because the client is Simple Start! So I quickly set that user to be the old original account, now with the new ID/email address, and then made that user the master admin for the client. Voila!
I now have separate logins for my firm as a company and my firm as a practice... and my firm is now a client of my firm. Seems very odd to me but is that basically how QBO is supposed to work?