I'm going to move this to the Kimber forum for you; you'll probably get more views there from folks knowledgeable on this issue.
There is a reference in Kimber's FAQ area about this, and it is strangely specific and vague at the same time:
Link: Kimber America | Kimber America | FAQ
"I'd like to purchase a .22 Conversion Kit; which model is it compatible with? What ammo works best with it?"
"Our .22 Conversion Kit will fit on our Full Size 5 in. and Pro Sized 4 in. barreled .45 cal 1911s. We recommend using CCI Mini Mag ammo for best function and reliability. The kits will fit on most, but not all, 1911 models."
They do specify .45 caliber 1911s only (pretty specific), and then they say the kits will fit on most, but not all, 1911 models (kind of vague).
I would recommend an email or call to customer support giving your specific model and caliber, and asking which kit (if any) will work on your frame.