Cat pee is strong stuff. I'm not sure there is any carpet material or treatment that will allow you to reliably clean a carpet in place. One thing that might work is replaceable carpet tile. At least one company (http://www.millikencarpet.com/americas/residential/carpetpanels/legato/Pages/default.aspx) makes carpet tile with a non-adhesive, sticky back similar to post-it notes. If you need to clean a tile or two, you pull them up and clean them by hand. You could probably perform a much more thourough cleaning and drying with the piece off the floor. If you keep extra tiles on hand, you can replace the soiled ones while you clean them.
I installed this tile in a basement bedroom because I was worried about flooding. Sure enough, we had a flood last year. A regular carpet would have been ruined, but I was able to gather up the carpet tiles, air dry them, and put them back down when the basement was dry again.
The tiles are good, with a nice texture and substantial padding, but not perfect. In some places, I can see the edges of the tiles where they join. They cost a bit more than regular carpet, even though you have to install them yourself. Lastly, you can't install them as an area rug; you have to do it wall-to-wall.
We bought the tiles at Home Depot, but I'm not sure they sell them there anymore.
Are you sure you want carpet at all? If the cat is going to mark your carpet 5% of the time, and a cat pees maybe 3 times a day, you're going to be dealing with cat pee in your carpet once a week. You say that hardwood is not an option, but there are a lot of laminate "wood" floors you could install that wouldn't absorb liquid at all. I don't think they cost more than carpet.