Okay, it's definitely not the tensioner.
I got all the parts in either Friday or Saturday (honestly, I don't remember which, all I remember is they showed up a lot earlier than I was expecting) and just installed everything today. New gaskets, new tensioner. The tensioner was a bitch to remove and replace because of the location and various bits and hoses in the way. I also had to use a dremel and shorten one of my allen keys since I have SW Motech crash guards installed and they just barely get in the way of getting at one of the side case bolts. Luckily I had a spare.
Anyway, I zip tied the chain as recommended and replaced the tensioner. The old tensioner appeared to be working fine with smooth operation when I was inspecting it. With the cover off and the new tensioner installed, I tried pushing on the chain guides to see if there was any extra slack. On the left run (tensioner side), it would not budge. On the right side run (exhaust header side) I could move the guide and chain about 2 mm. Other than that, I couldn't disassemble any further to inspect guides or anything.
With the cover off, I hit the starter button and the starter motor started spinning things but the engine never actually started. I'm thinking this is because of the PU sensor not being in the right place to pick up the sprocket. Anyway, with the starter motor spinning things, the right side run (exhaust header side) didn't move one bit. The left run (tensioner side) did vibrate some, though it was hard to tell how much. Maybe 5-10 mm or so? I don't quite remember.
Anyway, not being able to do anything more, I reinstalled the side cover and started up the bike. Lo and behold, it sounds exactly as it did before! I guess the noise isn't coming from the tensioner. To verify, I removed the top bolt on the tensioner and placed a small screwdriver in the hole. It wasn't vibrating at all, so I could tell there was enough tension on the chain. Either that or the plunger was pushing the guide to a point where it could not move any further and the slack was in the chain because of stretching. I experimented by turning the screwdriver to retract the plunger and immediately the noise got worse and the screwdriver started rattling. So now I know what too much slack feels and sounds like. There was a definite slapping noise and it seemed quite irregular. I then released the plunger and everything went back to the way it was before. I even tried turning the screwdriver the other way to possibly extend the plunger further and there was no change from what I can tell.
So, I guess my bike just makes a regular, periodic noise that sounds like a sewing machine. I walked away from the bike about 20-30 feet or so and could still hear the ticking noise. Honestly speaking, my engine has been running fine ever since I purchased it a couple of years ago. I've put over 16000 miles on it, commute every day, did two long distance road trips, and get great gas mileage. I typically get around 42-46 mpg commuting, but on my road trips I get upwards of 52 mpg. All that said, maybe there's nothing wrong with it and the noise is just my engine's character? Injector noise maybe?