The Nitto Ridge Grappler 285/75R17 is 33.8" x 11.2" by spec. The sidewall says 34" x 11.5". I did a quick left-to-right tape measurement at ~42psi and it seemed more like 33.2 to 33.3" to me when mounted. I don't feel or hear it hit the KDSS, and when parked under full lock there's about 1/2" of clearance - but there is a spot on the KDSS bar where the dust has rubbed off, so I assume under full lock and some motion (maybe reverse, maybe a right turn) the tire must rub the arm ever so slightly. For me it's not something I would worry about, but if you went with a larger tire you likely would need to deal with it. Keep in mind I'm running +25 offset wheels, so if you're running +40 you may have more rubbing.
On 35s I'm sure you'd need the KDSS relo kit. Frankly I can't imagine getting 35s under the LC without significant effort. Front bumper, serious lift, KDSS relo, and a major alignment effort. I know slee did it for @Markuson, but I can tell you that without pushing the caster significantly positive you're going to rub the frame at the rear side of the front tires when turning, not to mention all over the body under the mud flaps, the front bumpers, etc. I have maybe an inch of clearance between my tire and the frame, though the +25 offset wheels probably exascerbate the problem slightly due to the wider circumference of the wheel edge when turning.
It sounds like I would be better off trying to mount 35s on a stock LX570?