I was wondering a similar thing - if all cooked is no good and all raw is no good, what about partially cooked? Or mixing raw with cooked? I'm not an expert but I agree that raw bone is probably better than eggshell, since bone is closer to a cat's natural diet than eggshell. (said by someone who's not doing raw bone yet...)You're unfortunately having to throw darts at the problem since the canned food produces lose poop with no vomiting and the raw food produces solid poops but with nausea.
One of the things that seems to produce solid, firm stools in raw fed cats is the use of whole raw bone. When you cook the rabbit, you are using eggshell calcium instead of raw bone, and continue to see pudding poop. I don't want to experiment on your kitty like a lab rat , but I wonder what would happen if you served cooked meat with some small pieces of raw meaty bone for calcium? Just thinking out loud...
Our Laelia isn't as sensitive as your kitty, but I've noticed that the less I cook the food the more she pukes, and she's better with chunkier food instead of more pureed. Maybe just adding some chunks that will take longer to digest would help??