Hi Charity. My kitty Simba is diabetic (diagnosed 8/19), and was diagnosed with IBD about 3 years ago. Ultimately, his IBD was stabilized on a freeze-dried raw diet of rabbit and duck. He eats the food dry, not reconstituted with water. He was still vomiting some, but it had decreased a lot from pre-diagnosis. In May of this year I learned of a hair analysis test to identify food and environmental sensitivities and from it learned that there were other animal proteins he could safely eat, notably chicken, and that actually he was sensitive to duck, which I had been feeding him. I adjusted his diet accordingly, and now he has more variety, and seldom vomits at all. Up until August, when he was diagnosed diabetic, he was still eating the Royal Canin LID rabbit prescription food. Upon his diagnosis I removed that food completely, and searched for low/no protein dry foods. I found Young Again Zero food (zero carbs), and just recently found Wysong Epigen 90, which is 5.5% carbs. He really doesn't like the Young Again, so I give him a small amount of the Wysong daily, as a treat. His favorite treat is Whole Life freeze-dried chicken breast, which is cooked and then freeze dried.
What is disappointing to me, and puzzling, is that his diet for 3 years prior to his diabetes diagnosis was nearly zero carbs, as he ate no canned, only the freeze-dried raw. He never showed any symptoms of diabetes, so we were very surprised. However, he is too fat (16 lbs., should be about 10-11), so I am sure his weight was a factor. He is gradually losing weight now. Many people's cats go into remission by removing high-carb dry food alone, but he was already on a nearly perfect diet for diabetes, since he was eating very little of the Royal Canin dry rabbit.
Good luck with your kitty. I don't have experience with diet for lymphoma. There is a separate message board for it, though, which you might find helpful.
It does also contain pork though, so if he’s sensitive to that protein it may not be a good option