I also have a KK. With any non-smart trainer, you’ll have to shift as appropriate during your ramp test or any other workout. This is actually beneficial for riding with power on the road, because you are forced to learn how to shift to hit a desired power/cadence combo.
I would first pick your cadence (say, 90 rpms), then shift as required to hit the power target at that cadence. My ramp tests end in the 340-350 watts range, which on a KK means I need to be going 23-24 mph. So I start the test in something like my 39/19 gear, then shift down to the 39/14 or so, then doubleshift to 53/19, and my test generally ends when I’m at 53/14. I’m guessing on those exact gears, but generally I shift from my small ring to my big ring at some point during the test.
Again: Spin at 90 rpm or whatever, and shift as appropriate to hit the desired power number. I can usually find a gear that keeps my cadence withing 5 rpm of my desired cadence while hitting the desired power.
I used to have a dedicated tire, and it did slip a bit less than a normal tire. I’m not sure what it does to the power reported by the inride adapter… I have an older KK and use a crank-based powermeter (4iiii) to measure the power so I just use my old 23mm tires until they wear out. I have enough to last me several more seasons before I need to buy a dedicated trainer tire again.