You only wish it was the clutch. Clutch packs take an hour to change, and 40 minutes of that is soaking the new friction plates in oil.
If you are very lucky, your shifter shaft is shot, or, more appropriately, the little bit on the end that engages the star on the end of the shift drum is. That bit usually isn’t serviceable, though, so you end up having to replace the shift shaft. That will give the “empty” non-shifting pedal presses you describe. I’m not intimately familiar with your generation of ZX-6, but the shift shaft on my FZ-1 took about 30 minutes to change; it hides behind the clutch pack. A pro mechanic is going to charge you for the part and a full hour’s labor. If it is the problem, the fix is going to cost you about $150ish. I could usually still ride by finessing the transmission in various ways, double clutching, playing the throttle, all whilst jamming on the pedal until it cooperated. The bad thing is you’re seeing it in particular gears; mine was through all 6 and ONLY acted up on downshifts.
Sooooo chances are your problem is in the shift drum itself, or perhaps the shift forks are shot. If so, you’re going to be splitting the case, and that means a grand minimum for labor only. Cross your fingers, yo.