I found another way to add your account without disabling login approvals. Hopefully it can help someone.
You need to have Code Generator set up on your iOS device (in the Facebook app, in the left-side menu, tap Code Generator).
- On your Mac, try to add your Facebook account using your normal email address and password.
- You will get an error message telling you that either of those were invalid.
- Right then, launch the Facebook app on your iPhone and tap Code Generator.
- Back on your Mac, try to add your Facebook account again, this time using the six-digit code provided by Code Generator in place of your password. It will work.
Don't skip the first step. Even though the login fails with your normal password, it seems to be a necessary step in my experience.