I don't think you're going to find a climbing carabiner that small. The ones I have (not that I climb any more!) are more like 4 inches for the smaller non-locking ones. The metal diameter is a little less than your pinky finger. You would have to put a 1/2" (maybe 3/4") split-ring on the key and then clip the split-ring to the biner.
Have you thought about something like this instead of a biner?
Yeah, I don't think a "real" biner can be found much under 3". I do have one of those cable rings; it's not bad but not exactly what I need.