Hi there,
If this were a human, a vegetarian diet would be the way to go. You can look at this option for dogs, too, although I think it would be important to find a real food vegetarian diet. Not all dogs who excrete high levels of cystine form stones, and it may be that the stones also need a nidus to form around. It's my theory that renal tubule cells damaged by low grade inflammation provide those nidi. Studies have shown that renal inflammation happens pretty quickly on consumption of processed food meals in dogs. Moving to real food may reduce nidus formation and thus cystine urolithiasis, regardless of actual cystine levels in the diet. There are a lot of 'ifs' in this paragraph, but clinically I find most urolithiasis cases fit this general understanding in how they behave clinically.
The other advantage of a vegetarian diet is that it alkalinizes the urine, helping to solubilize cystine. If you use soy as the protein source, then it contributes phosphates that reduce cystine clearance. Sweet potatoes as a carb source have almost no cystine, so overall cystine levels in the dog should plummet with this approach.
Hope this helps!
Steve
Steve