I'm not sure if they sell IR specific cameras with all of those requirements you've listed, and I'm almost certain they would be more expensive than you would want if they did.
Another option would be to purchase a webcam and hack it to only see IR light by putting filter in front of the lens similar to the lens on the wiimote, then run the registered image through a tracking code to see where the bright spots are. I know there are sites on the internet that specialize in this.
The problem with going with a webcam is that the tracking resolution will be a lot lower if you're staying within a small budget. Also, the lag will be greater on a webcam if you go with a lower end one.
I personally have not dissected a wiimote and hooked it to a microcontroller, but I know there are some really good tutorials out there that I've skimmed through. If you end up going in that direction, we'll help out in any way we can. Good luck with your research and with your project.