Your suspicions may be correct regarding the possibility of a higher output cartridge, but obviously I have no clue about that, so I won't bother to address that part of your post. (Good luck with that if you're unfortunately correct!)
However, before giving up, I would change the resisters on the phono card to decrease the gain in the K-1xe. (This will push the volume control higher for your phono section, up closer to the 12 o'clock position that you like.) I don't know if you realize it, but when Ayre did the Evolution upgrade to the K-1x, it actually increased the maximum gain from 60 (as listed in the manual), to 70 db. (In case you are wondering, I got this information directly from Charles Hansen himself via a Audio Asylum post a couple of years ago). It can't hurt, and who knows, maybe it will solve your problem.
Now, with that being said, I recently upgraded my Shelter 90X (0.60 mv output), to a Dynavector XV-1S (0.30 mv output). I tend to listen at pretty realistic sound levels, and with the Shelter, my volume control was typically around the 9:00 position. However, after the change to the Dynavector, I typically run around 10:00 to 10:30. (So, it does sound like you might have a cartridge with 0.60 - 0.80 mv output.)
Good Luck, and hang in there!
However, before giving up, I would change the resisters on the phono card to decrease the gain in the K-1xe. (This will push the volume control higher for your phono section, up closer to the 12 o'clock position that you like.) I don't know if you realize it, but when Ayre did the Evolution upgrade to the K-1x, it actually increased the maximum gain from 60 (as listed in the manual), to 70 db. (In case you are wondering, I got this information directly from Charles Hansen himself via a Audio Asylum post a couple of years ago). It can't hurt, and who knows, maybe it will solve your problem.
Now, with that being said, I recently upgraded my Shelter 90X (0.60 mv output), to a Dynavector XV-1S (0.30 mv output). I tend to listen at pretty realistic sound levels, and with the Shelter, my volume control was typically around the 9:00 position. However, after the change to the Dynavector, I typically run around 10:00 to 10:30. (So, it does sound like you might have a cartridge with 0.60 - 0.80 mv output.)
Good Luck, and hang in there!