The TKC80s are more of an ADV tire rather than a dirt bike or DS tire. I have them on my CB500X, but have never had issues like you describe, even taking the rear below 20 PSI. I think others are on the right track with the different rubber and the narrowness. The lighter weight of the KLX may also be a factor.
On a bike with pizza cutters, it needs to be going faster to be stable on sand and such, like above 20 MPH or more. Like a boat, you kind of got to get it up on a plane and float near the surface. The wider tires of the TW can accomplish this at slower speeds. I think you've got to grip it and rip it a bit. I would also have a digital tire pressure gauge and a pump and I would run the same mile or so back and forth dropping or raising the PSI 1-2 lbs. at a time noting any differences and finding the sweet spot. It seems like most can run the rear tire about 3-5 PSI less than the skinnier front tire.
I got the TKC80s for the X because of the 17" rims front and rear. There are limited aggressive tire options for a 17" front. The 17" front, even down from just the 18" of the TW is also going to make a difference in handling. I do love my X for the ability to go further and faster in a day, excel on pavement or give me the option to go on a dirt road or double track trail, but I prefer to ride the TW whenever I know I'm going to be offroad. It's just easier and more comfortable to ride. I've had that X on the White Rim Trail and other places I probably shouldn't have taken it. It'll do it, but I'd rather be on the dub.
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