2 Pack, Light Socket to Plug Adapter, Convert Light Bulb Socket to Outlet Adapter (White).

It really isn't that hard to test. You can get one of those light socket adapters (like one of these things, you can get them in the light bulb section of your local hardware store) and replace one bulb in a light with multiple bulbs with your WiFi router's power cord. Turn on the lights, wait for it to connect, start a video call, get the other person to record it, turn off the light, and then watch the recording after taking it all down and putting your house back to normal.

So connect another light to the integrated light? If it uses standard light bulb sockets, you can get an adapter for those to plug in some other light fixture. Something like this; https://www.amazon.com/Light-Socket-Adapter-Converter-Outlet/dp/B0B24TX1QG/