I set up my Private Mode with my fingerprint, but when there was a new update, it's now making me enter with a password which I never set. Is it possible to use my Private Mode again without a factory reset?
There's no way to "get around" a password - on anything. You have to remember it. If you can't, and you remember your Google login and password, you can factory reset your phone (you'll have to enter your Google password) and start again. If you can['t remember your Google password, the phone is useless.