Hi @emmaandchubbs We are on a raw diet and I learned a couple things while researching the change from kibble to raw. Most importantly, raw food and kibble/cooked food will digest very differently and should not be combined in the same meal. Much like people, different foods will digest differently and if in poor combination, as is raw/kibble, can cause digestion problems, vomitting and gas. Penelope is extreme in her sensitive tummy, for example, if she eats her raw meal, and then finds cat food kibble and eats even a small amount, it can make her sick.
I do not know how a bully tummy would respond to alternate kibble, raw meals, but both my vet and the food expert at our pet food store advised to switch completely at one time and not to ease in the new food as one would with a change to kibble.
I am sure someone with more knowledge will come along soon for you. Good luck!