Installed a new alternator and now get a message that says "battery not charging"

The first part of the test involves measuring battery voltage with the engine running, which you already observed is falling. The third cause I eluded to is a high-resistance break in the wire going from the generator's output terminal back to the battery's positive post. One test is to measure the voltage at that output terminal when the engine is off. If must have full battery voltage all the time, but that can be misleading. If you find 0 volts, there's a burned-open fuse link wire or a blown very large fuse that is typically bolted into the fuse box.

https://www.2carpros.com/articles/how-to-check-a-car-alternator

It's when you DO find voltage that you have to look further to identify if there's a high-resistance break. For example, when fuse link wires are used and it burns open, it leaves a carbon track behind. That carbon will conduct enough current for a voltmeter to pick up battery voltage, but the generator's output current will never get back through there to charge the battery. The clue to that is with the engine running, you will find considerably higher voltage on the generator's output terminal than across the battery terminals.