You're lucky you are able to overcome your lifelong allergies. I grew up with a cat and was fine with them. As soon as I left regular access I got allergic. I don't know that it's the norm to be able to overcome allergies as well as you have. For those that want a cat regardless of allergies, I've also heard that the shots can help a lot.
I'm not sure it's fair to say, though, that certain members of a family should just suffer until tolerance is built. It also might not be fair to the cat because sometimes with long exposure, allergies can get worse and then the cat would have to get another home.
In my experience, fostering rarely lasts one week. In a story for another time I actually fostered a cat that was then dumped on me for years, but this was a terribly run organization and they have since undergone new management. It is more likely that a foster cat would be with the family for at least a couple months until a "forever" home for it can be found. It could easily take kids that long to realize they are or are not up for scooping that much litter.
I grew up with pets and love them, but if it doesn't work for the whole family it's going to be a problem. If the question by the OP was "We want to get our kids a pet but wife is allergic" I would suggest guinea pigs. I hear they are wonderful with kids, and since they mostly stay in cages it is much easier to limit allergen exposure. They do not, however help with mice.