... because I'm shooting flames from the exhaust on a regular basis.

I traded a Suzuki 2005 GZ250 for a low-mileage 2008 S40 that already had a Harley Davidson pipe on it. I'm not sure of the type yet, but I'm researching. I LOVE the sound of it, but I'm pretty certain it's running too rich. I know about the frequent backfires on deceleration with the S40, but I'm also shooting blue and sometimes yellow flames from the exhaust when it backfires. There's sometimes a blue puff right after I shut off the engine, and if I jet down the interstate for awhile and then turn into an offramp I'll have a nice Batmobile jet on the way off.

I've seen tons of carb-adjustment threads so I'm looking at those. I've tried removing the airbox and filter to see if I could change the mix a little without tearing into the carb yet. I occasionally get a little soot coming from the exhaust but not much.

I'd love to keep this pipe, but I'm willing to lose it if I can't get it working without the dragonbreathing. I'll have pics of the whole thing up in the next 24-48 hours if that helps. Any information is appreciated. I'm really loving the Suzuki so far.

...and,... location, location, location...
Jetting is entirely dependent on where you are and where you ride...
Humidity, temperature, and elevation...

Okay,... lets say,.. Arkansas, nearly flat, nearly sea level, (less than 2000ft)... with a slightly open muffler,... (open the airflow to match, if it hasn't been done already)... k/n filter, or oiled foam.
#152.5, or #155 main jet, #55 pilot... 2/3 to 1/2 thick needle jet spacer...
Unless you are radically different than that (see 6's, ie, #160+main/#60+pilot)... look for a leaky petcock (very common),... or stuck float, or stuck needle slide (fairly common)....

...or,... just learn to love flames.... (shoot!,.. some people would pay for those!).... big singles, are poppy, and sputtery,.... maybe a little flamey?...
Lets just say primitive,... Cool!... ...

I live in Oklahoma with the same climate as the OP. Mine does the same thing with the flames. I tried a #55 idle jet instead of #52.5 OEM and a #150 main jet instead of #145 OEM and the bike ran like crap. Way too rich at idle, with black smoke sometimes and occasional cut outs at WOT with black smoke. That was with a new Iridium plug. I tried to make some adjustments, but didn't help. The bike seemed to have less power at WOT. It has the OEM air filter, which was fairly new. The OEM muffler has the baffle removed. So I chnged it back to the original jets and just live with the bangs. BTW, I have no exhaust leaks.

Sounds like you are running too good for it to be the petcock....
Gotta' be rich on the jetting... since you bought it with the Harley muffler, chances are the jetting is not stock...

Cool!... Flames!!.. ...

I'm not too impressed. Way back in 1960 I saw even more impressive flames jumping forth from the exhaust pipe on a Cushman Eagle. Another of my acquaintances removed all but the manifold from his 3-horse Cushman, brazed an old air horn bell onto a steel lawn chair leg (to get the ninety-degree bend) and stuck the resulting pipe into the Cushman's manifold. On deceleration it would throw basket-ball sized fireballs. It was loud enough you didn't really want to follow him very closely.

Mine looks the same way at night.

First off, these carbs are Altitude Compensating. At a higher altitude there is a lower  baro pressure and consequently the needle and slide do not go up as high so the ratio of air/fuel (14 to 1) stays close to the same.  Caused by the pressure differential between the top and bottom of the carb diaphram. So I have been from sea-level to 5000 ft passes without any adjustments and it just runs great at any altitude.
We run these carbs on small aircrafts from sea-level to 15000 ft with NO adjustments. Strombergs, SU's and our Mikuni's do not need to be fiddled with.
Now a standard carb like the old Bings (and Holley's on your car) do need to be adjusted for altitude. See my setup is below and my exhaust does not have any blue-ing. It looks new.