A couple possibilities...
1) That the position of the freehub body on the KICKR is slightly offset from the position on your wheel.
2) The spacing on the cassette that came with the KICKR is different from (I assum your) Shimano's spacing.

Try putting your road cassette on the KICKR and try that. If it shifts better, then you know the problem is the cheap KICKR cassette. If it still shifts poorly, then there is something wonky with the alignment of the cassette body on either your road wheel or the KICKR.

I bet it is a No1.
Alwasy when I put my bike on trainer (direct drive) I have to trim rear derailleur, then putting the wheel back, trim it back. With the electronic shifting at lest with the e-tap it is very easy and quick.

I have a wheel on trainer. Always have trouble getting into 11 on the trainer, but no issues on the road (granted I'm rarely using 11 on the road, and I just stay away from it on the trainer now). Brought it up to the mechanic at my LBS when I first started riding and he said it was common. I've just assumed since then that there was some kind of force from the clamp while on the trainer that made that final shift difficult.

Swap the cassette, mine would shift like garbage moving up the last 3-4 cogs and really heavy feeling moving down to the 13-11 range while being flawless on the road. Swapped to a 105 cassette and shifts as smoothly as my road wheel now.

Another possibility is that the clamping force of the Kickr quick release slightly pushes the derailleur hanger and hence rear mech to tilt slightly inwards. I have noticed this happen on my Lemond Rev.

Listen to the TrainerRoad podcast this week. They answered this exact question from a user. I don't remember the answer because I don't have a Kickr

most likely its the offset of the cassette from the dropout on the wheel vs the kickr. just adjust the derailleur cable tension. (or whatever you have to do with electronic if you have that)

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double check the 11 speed on the kickr..

My Kickr had the same problem. Turned out there is a shim that wears out on the freehub. Once it wears down, it throws out the alignment. Talk to Wahoo support about getting a new one.

Watch this video for more information: https://www.youtube.com/watch?v=8eeUiZ4H7kg

This is certainly a timely thread. Just got my Kickr set up a few days ago and I'm having the same issue. Guess I'll try the easier options this weekend.