The hardware version is v1 or v1.2?
How about status of all lights on it after you finish the configuration?
Hard reset it and login the web interface again. On its status page, check firmware version of the extender.
Upgrade the latest firmware and reconfigure it.

When I press finish, it hangs. I then get a message to either reconnect to network or to try again. The Extender disappears from the list.

It is normal the extender disappears from the list when you finish the last step. The range extender will copy the wifi name(SSID) and password from your main router. So when it is configured, the original TP-LINK Wi-Fi will no longer show because it has changed to the same name as your main router's Wi-Fi.

Check the RE light on the unit, If the RE light shows up, it means the configuration is done. Just wait for 1.5 minutes, when you see RE light shows on and the single LED (half circle LEDs) is on, setup is successful. You will see only one SSID in the list but with stronger signal.