Check to see what's on the *inside* of the wall you're thinking of mounting your Optex VX-402R. Is there another hard-wired PIR in the vicinity?? How about an electrical outlet?? You mentioned that you had no power where you were thinking of mounting the sensor. Have another "look-see" and determine whether-or-not you can obtain the coverage you need if you moved it closer to an adjacent interior device or outlet. Don't limit your "vision" to just the lower floor either. If there's something on the second floor you can exploit, go for it.
Personally, I would secure your pet door for the evening (and contact it so that when you arm your system, you're arming the door). IMO that would be better than using a motion detector which will pick up all sorts of extraneous movement (like bushes and trees blowing in the wind, other animals, or even people). Keep the protection *on the door* and you'll have far less headaches (and probably sleep better too).
The alternative is to simply contact the pet door using your 5816 and have it beep, chime, or turn on the lights (instead of the motion detector). I guarantee that you'll opt to secure the door after a few "incidents" and you won't be out the price of the Optex unit if you decide to "scrap" that idea.