Electrical – tap into an exterior light in order to power a security camera

Your LED security lights work on PIR, passive infrared to trigger them on.
The Lorex Camera specs out using
Night Vision
Night Vision Distance 41m
Number of Infra-Red LEDs 24
It is possible the infrared of the cameras are over loading the security lights enough to keep them from working fully. You can check this simply by turning off the camera or placing a piece of cardboard over the front and see if the lights trigger as expected.
If the two interfere with each other not much can be done other then moving the camera or turning off the night vision ability (kind of defeats the use).
It might be possible to make a small shield to shade the cameras IR from directing into the lights or a directed tube on the lights sensor to narrow the view.
If the interference exists, it is all trial and error to find a placement or shadowing to make it work. My drive camera had to be raised about ten feet above the leds to make things work as best possible.

What you can do is cheat and drill a hole through both the interior and exterior walls. You'll need an extra long drill bit and make sure there's no electrical in the way (i.e. identify where the studs are and where the outlets are).

Be sure to caulk the outside hole with an exterior grade caulk