If it is allergies, it will probably be tough to find a single protein source food that is also low-carb. I know there are different schools of thought on this (and honestly I don't know how much of it I believe myself) but my vet always said you need to do a several-week feeding trial of a novel protein to see if it helps with potential allergy issues. The novel protein just means it would be something the cat has never eaten before. (So, probably not chicken or beef for most cats.) The problem with this is that even "allergy" foods or foods touted as being for sensitive stomachs, etc. tend to contain some chicken, or if they don't (e.g. Natural Balance, Blue Basics, or prescription allergy diets), they can be high-carb. For example I got all excited about Instinct duck flavor and then it turned out to have chicken in it. Blue Wilderness Duck is, I believe, a variety that is both low-carb and single/novel protein, but it's expensive and pretty high in calories and fat. Of course if the budget allows, you could always try it to see if your cat tolerates it, and just make sure you control the portion size as needed. It's at least pretty widely available. Another brand with several truly weird protein options is called Addiction, but it's only available online for the most part, is insanely expensive, and my cat didn't really care for it anyway.
Other people here have said that they think a raw diet might be the best for allergies. I'm not sure why this is but it's something I have heard repeatedly.
If the budget and logistics allow, I think the easiest way of accomplishing all this might be to try a raw pre-made frozen diet. Primal makes raw nuggets in a number of unusual proteins, or at least they used to (venison, duck, etc.) and I seem to recall they are acceptably low-carb. Of course they do contain some veggies and other ingredients, so if the allergy is not to a protein, this could be a drawback between a pre-made vs. a homemade raw diet, where you can completely control what goes into it. I would like to feed raw, but I'm pregnant and will soon (if all goes well, of course) have an infant in the house, so these are situations where they say raw can be a bad idea.
Again there are many schools of thought on allergies so I'm not saying this approach is the be-all end-all, or anything.
Good luck! Allergies are such a pain.
