The Russian made Red Rod, at 46mm is the shortest... but no clone, so at $75 before shipping (last price I recall), is a tiny bit out of your price point.
There are tons of Tarsius 4Nine clones, which are of course, 49mm. Of the ones
listed here on FT... I'd suggest the $20 and over versions, as the quality is much better than the $15 ones.
As they are a close tolerance, hybrid type mech design with a floating button and no positive pin, quality is critical to good performance. These same mechs are being sold in the US for $40 and up.
I've seem to recall seeing a 44 or 45mm mech... but can't recall what or where. ;-)