If I’m understanding correctly, the lock you’re facing is not a lock on the SIM card, a screen lock, protecting your phone from use if it were lost.
When you get a new phone, if you try to move content that has been backed up by Google (old text messages, call history, apps and app data, saved WiFi networks) to the new phone, the new phone will require the same PIN before that data will be moved to it.
Whether or not you can move the SIM card has nothing to do with the screen lock. It has everything to do with what kind of SIM card is in the phone, and whether or not you buy a new phone that is compatible with Republic Wireless. So no, moving the SIM card will not require that screen PIN, but the Community cannot tell you whether or not your SIM card can be moved, because the Community does not know whether you have a GSM SIM card (which can be moved to a new, compatible phone) or a CDMA SIM card (which cannot.)
@grandbobby pointed you to information that will help you determine whether or not your SIM card can be moved. Unlike other participants here in our Member Community, I can look at your account and tell you whether the SIM card can be moved, but I’d need your permission to share this account-specific information publicly.
More to @grandbobby’s point, factory resetting your existing phone may restore your use of the phone (without having to buy a new phone in a crunch), but you’ll have to know how to sign in to the Google account on the phone in order to get past the PIN lock once the phone has been factory reset.