It’s unclear if you have already registered the domain “thebusinessname.com”, you do state that you “want” to register it, so I’m assuming that you haven’t.
When you enter an email address into a facebook email field, it attempts to “test” the email address. If the domain is not registered, it tell you immediately that the email address is not valid, and won’t allow you to save. If the domain is valid it sends a “test” email to that address, and until you click the link contained within that email facebook treats the email address as unverified and will not display it to anyone other than yourself.
The reason you can enter the other email address is likely because that domain does exist (even tho it’s not your domain).
I’m afraid you will need to register the domain first.