I agree that you might find it. If you wife is more attuned to the cat and its habits, she might be able to find it given a little time. If you decide to have it replaced, a pipe repairman could cut and fit a stem for about thirty or forty bucks, so compare that with what it would cost just to buy a new pipe. If you get a new pipe, save the briar from the other ... I think the stem will turn up during house cleaning. Cats hide their treasures, but usually in fairly handy places, from their point of view.
My cat likes to pick up items. He took a key out of our buffet, but was so kind as to leave it in my shoe, as a present. He also leaves me his cloth mice if I behave well. Hey, who is domesticating whom?