Apple doesn't sell the official battery as a standalone part even to Authorized Apple repair shops, so anyone telling you they're installing an official battery is already lying to you or putting in something used.

Even if a local shop somehow sourced a legit, new Apple battery, why wouldn't I go to the Apple Store if it's the same cost and would only be the battery?

(For $299, Apple replaces the speakers, touchpad, batteries, top case, and keyboard and provides a parts and labor warranty for 90 days)