Change iOS account on iPhone without losing data
You cannot add the business account for 2FA to your iPhone without first logging off the private account. However, when you have logged into the business account and enabled 2FA, you can logout and then login to your private account again.

Instead of purchasing a new phone for the business, you can also use any other modern iOS or macOS device for enabling 2FA. I.e. you can use for example a Mac computer or an iPad or iPod Touch.