The problem with trying to use any “canned” curve with any alternative process is that there are so many variables involved that the chance the IJM supplied curves will work for you are fairly slim even if you were using the process that the curve was intended for. As we have been discovering, even the lightsource plays a large role. You are using a curve made for a different process which makes it next to impossible. And Kallitype is highly variable itself.
Do you have a scanner? If so, try using that. I think there are instructions in the manual. There is a 21-step tool especially for that called 21Step_Scanner_Tool.
What are you evaluating by eye, an actual image or a text target? If you already trying to print images, you have put the cart before the horse. If you are working with a test target, how are you modifying it? I think you actually ought to be able to get something that prints full-range doing this by eye, even though it will not likely be linear, but you have to evaluate it correctly and make the appropriate adjustments to the curve. For this you would use the Piezography_Curve_Adjustment tool. Since your highlights are too dense in the negative, you would shape the curve so that there will be less density in the highlights by pulling down that part of the curve. This is basically the Limiting process described in the manual.
Whatever you do, figure out all of the Kallitype printing parameters that you want to use before starting to adjust the curve, because if you change any of those mid-stream you will need to start over.