Problem with cassette and derailleur

You can't use an 8 speed shifter with a 10 speed cassette. The space between individual cogs is smaller than on an 8 speed cassette. The shifter moves the derailleur a fixed distance per shift, so it can't reach the position of the cogs if they are not spaced as intended.

The fact that you can use the 4 biggest and the smallest gear is pure luck, because the derailleur seems to match those positions good enough to make it work.

In conclusion, try to find an 8 speed cassette that fits on your trainer.

This answer stems from comments on @Max's answer: if you intend to only do structured workouts, then if you can find a gear where there's no chain rub, you can set the trainer to erg mode. In erg mode, the trainer automatically adjusts its resistance. You don't need to shift. In practice, even on erg mode workouts, I do find myself shifting when I need to change cadence markedly. Otherwise, you would just change your leg speed - I find that when decreasing cadence, it takes a while for the trainer to catch up to me, whereas increasing cadence is easier.

I would view this as a short-term solution. You are stuck to one gear. If you want to do something like group or solo rides outside of structured workouts on Zwift, RGT, or Rouvy, you will want to shift. The clearly better solution is to get an 8s cassette. It will fit on the freehub body. Take note that the Kickr should have come with a cassette spacer, a small metal ring. (At least, this is what I believe I received with my Kickr.) Do not discard it. You will need it to make the 8s cassette work, you put it on before mounting the cassette.

As already discussed, your shifters and derailleurs are designed for 8 speeds. The shifters have 8 different positions. There are only 7 'clicks' in that shifter. You can't remanufacture that shifter to shift through 10 cogs. Well, maybe you could, but this requires the resources of a large manufacturing company like Shimano. The internals of Shimano's shifters are very complex, and they can't really be disassembled by hand.

As a side note, there are a handful of edge cases where you can upgrade your shifters by one speed. Campagnolo's shifter internals are simpler than Shimano, and the 9s and 10s shifters had a ratchet ring inside that controlled how much cable the shifter pulled. You could simply order a 10s ring from Campagnolo and put it in a shifter. Relative to home mechanic skills, I'd characterize that as a medium to high complexity task (and I paid a shop to upgrade a pair of 9s shifters I had). In that case, you didn't even need to change the rear derailleur. In the case of SRAM 11s shifters, Ratio Technology 3D printed a ratchet ring and some other parts that take the unit to 12s, but this does need a new rear derailleur. Again, in Shimano's case, the internals of the shifter are much more complex, so I don't expect a third party solution to enable you to convert between speeds.