Have you tried other guitars? This is not a common issue (never heard of it before), I would want to check other guitars.
Yes, I would think that some form of pass filter or shelf would take care of the issue, since to my understanding, Presence is a boost to the high shelf. Generally a Presence knob can only boost high freqs, but not cut them. I would not be surprised at all to find that the KPA can cut them, which makes the whole thing even stranger to me in a way (because you like it at a negative value).
Yes, printing the effects is not a great move for several reasons. Obviously, once they're printed, that's it, no adjustment can be done, and effects act differently in a mix than with just guitar playing. The effects on the Kemper are fantastic. For an amp. They are pretty poor when put up against any good VST in your engineer's DAW. Like you noted, they are printed in mono, and there are a lot of cool things that can be done in stereo.
Yes, always record a DI. Not a bad idea to generally take Nicky's advice pretty seriously.
I don't know my way around your interface/routing software/DAW, but I think this is definitely possible. What is the output level of the KPA set to when you record? What is the input of your DAW set to? My Babyface will not go below +9db, and I leave it right there, as it should be for a line level signal. You would turn the Input gain up if you were recording through a mic for example, but with my setup at least, turning up the input gain on my interface is a bad idea.
What do your waveforms look like within your DAW? Do they fill the whole verticle section, or are they much smaller? What db level do you record at?
Edit: Paul's post below sounds pretty dead-on to me.