The stick in yer TE is a pretty nice Sanwa, JLF-TP-8YT-SK-W if I remember correctly. The clicking is indeed a positive sign: that's a high quality stick.
I don't think you can get a happ/american style stick in there without some serious modding work, but if you're set on having a quiet stick, the best one on the market is the
Happ Perfect 360. It doesn't rely on microswitches. Instead, it relies on photo electronic switches.
Finding a decent tutorial for wiring the stick is difficult. I'd google that bad boy if you decide to make it happen.
Here's a write up of wiring the P360 to an older controller. Might give you a basic idea.