I think your best bet is to set up a 'cat cam'. We did this once when a few of our cats were fighting. We wanted to know who was the aggressor and since it was only happening whenever we left the room, we set up our camera and found out that our little cat 'Tiny' was actually the one starting the fights!
At that time we used our 8mm camera. The tapes for those cameras are usually 2 hours but you can get them for longer periods of time. It sounds like the cats tend to pee on the runner the most so maybe you could set the camera facing this area and then leave the house. Come back and watch the tape and you might have your culprit.
If you don't have a video camera, you could rent one or you could use a web cam. We set one up at work because money was going missing and we wanted to find out who was doing it afterhours. The web cam cost about 30.00 at Staples and was easy to install. It is motion activated and if you put it on the highest sensitivity setting it will make sure that it picks up the cats movement. The only problem would be to have the computer near enough to the spot they pee in because the cord is only about 3 feet long.

I think the kitty prozac is a good idea once you know who it is. Medusa has used it with great success with her cats for this very reason. It would certainly be worth a try before taking the more serious step of rehoming them. Hopefully it will all work out with a happy ending.
Good luck with the detective work! Let us know how the 'sting' goes.