I just checked the label of the Canidae Chicken & Rice formula, and there are no potatoes of any kind in any form in this particular one. However, there are peas, which are the 5th ingredient on the list, so that puts you back to the question of just how sensitive is he to peas. As someone mentioned earlier, perhaps the sensitivity level is extremely low, and a small amount may not trigger an allergic response.
Since you dont have issues with salmon you may have more options. Perhaps a trip to the pet store (Pet Food Express carries a wide variety of better quality foods, and they have a really good selection of each brand to choose from, so it is a great starting point to save on running around if you have one in your area) for some label examination would give you some additional options. Keep in mind that just because one product in a brand contains certain ingredients does not mean all of the products from the same brand do. As I said, the only Canidae dry food with no fish or salmon is the chicken & rice formula, and the only Natural balance products without those are the sweet potato & venison and their vegetarian fomula. We mix the three foods in a ratio that keeps everyones stools normal, as a couple of the pack find just the Canidae alone too rich and end up with really loose stools, but we didn't want to give up a better food choice because of it. It's all about finding the best you can that works for your dog(s), and sometimes that means you land a little lower on the quality scale than you might like.
A note on the Soloxine: as with any medication to control a condition, dosing depends on the severity of the problem as well as the individual receiving it, and there is a period of adjustment to find the right dosage to level out the condition. In Tenaia's case, when she was initially diagnosed she was put on 0.8mg once a day and was retested at 30 days to see where her levels were. I was told this was a normal starting point, & though I don't know how true that is, it was the dosage that worked for her situation. She is 12.5 years old now, and her levels changed as she aged, so she is now taking 1.2mg per day, which I split into one pill in the morning and 1/2 a pill at night so she has more of a constant level throughout the day rather than a spike and a dip, as she seemed to get shakey in the evening when she was dosed only in the morning. We also give the meds with a "snack" (about 1/3 of her normal meal) in the morning so that she has some food in her tummy (she is on Rimadyl as well for her arthritis) and so I don't have to cram pills down her throat - I hate doing that & she is really good at eating meds in her food.
I hope that is helpful. Good luck and let us know how it goes.