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Of all the stones you have the thuringian might work. I have not tried a C12k on kamisori, but if it does not work you will need a proper Jnat.

If you have magnification, do a few light swipes on the coticule and look at the edge, you should be able to tell how much work the bevel needs.

I use edge leading like any other razor

My coticule is 40mm wide, I keep the kamisori @ ~45 deg to the stone so that the blade fits on the surface, and do up down strokes.
For kamisori it is also very important to set the bevel on the...

usually on kamisori you do not want to go under 4-5k unless you have to fix some serious issue.
In your case the 4k is the lowest you should go, I personally use Coticule for bevel setting on...