I'm in a similar position. I have one "personal" Apple Id that I use for my everyday life. This uses my personal (.mac) email address.
I have a second Apple id that is my company email address that is used to submit apps for my company, and a 3rd Apple id for a client company.
Only the personal Apple id is ever used to log onto an Apple device.
How on earth am I supposed to handle this situation?
You can add aditional iCloud accounts to a device of your choice. Turn off all the switches so nothing is synced from that account. From then on login attempts using 2FA will prompt with the code on that device.
Settings -> Passwords & Accounts -> Add Account
I had the same issue, did some research, called Apple and ended up by writing a small summary at -> https://medium.com/@jelledelaender/2fa-on-your-second-apple-developer-account-46cec91838f0
I hope this will help you! It was the same issue I had. Having a not-primary Apple ID for development, where I couldn't activate 2FA on.