When I'm setting my Apple Airport Extreme I can't get out of the loop of "Updating"

Have you tried resetting the Extreme yet? If not, I suggest that you perform a "factory default" reset on it to get it back to its "out-of-the-box" configuration so that you can set it up again.

To do so: Start with the base station disconnected from power. While holding in the reset button, re-apply power to the base station. Keep holding in the reset button until the status LED starts flashing rapidly, and then, let go of the button.

Let's start with which exact model is your AirPort Extreme base station? You mention that your goal with it is to extend the Wi-Fi signal of your Wyze base station. Will the connection between these two stations be wireless or wired? The reason I ask is that in order to extend the range of another wireless router with the Extreme via a wireless connection between them, that other station must be another Apple base station.

A few things stand out from the AirPort Utility image you provided:

• You are using the AirPort Utility on a Windows PC ... correct? Which version of Windows is it running?
• The version of the AirPort Utility for Windows was designed to work with Windows 7. Apple has not updated since. It works best with the earlier version of the base stations, but may be problematic with the 802.11ac model, which you have.
• The IP addresses you show in the provided images, is outside of the local IP address range. Those addresses would start with ether 10, 172, or 192.

Do you happen to have an iPhone? If so, you may find it easier to use the AirPort Utility that is included with the iPhone.