There’s a similar topic:
https://forum.audacityteam.org/t/changing-the-click-track-sound/30599/1
It wouldn’t be that hard to allow custom tones in the click track generator (or rather a clone of it).
The nice thing about generated clicks is that they do work for any sample rate. Stored Audio has to be pitch-shifted in order to abide by this constraint.
The simplest solution is in your case to add a fourth sound choice “Sticks” and to include the samples directly (i.e. as numbers) into the code.