Hypothyroidism and diet

My elderly pooch recently began displaying some odd behavioral changes, which I may have had a hair-trigger response for because I am a behaviorist by profession (human, not animal) and because I have a rare disease and My dog was having behavior changes and after running tests, it turns out he has hypothyroidism (as well as a UTI, but that should be cleared up pretty soon).  My question is whether I should be more concerned about his diet now.  Being on Social Security Disability (and thus food stamps) myself, I don't have a huge budget for my two dogs, but I don't skimp on their needs either.  I can't afford anything more expensive than the brands they carry at the grocery store as far as dog food goes (unless it's a short stint on vet diet stuff for a kidney infection or something), but if it will be better for my little guy, I could probably swing upgrading from the regular formulation of adult dog food to either the "senior" style or lamb and rice formula.  Both my dogs currently eat the regular adult Pedigree.  Is there a particular diet that is more beneficial for dogs with thyroid problems?  Would there be a big difference to him between the different formulations within the "grocery store" quality lines?  I suppose in an ideal world I'd be feeding the dogs one of the pricier pet store brands, but it's more important right now that I be able to afford each of their medications and supplements (my other dog has her own health issues as well).  Oh, I should mention in case it is relevant that we also recently started my hypothyroid dog on probiotic dog supplements.

Thank you for any advice you can give.

[Aleda M Cheng, D.V.M., C.V.A]

The following is a list of good pet food store brands:  Merrick's, Spots Stew, Natures Variety, Wellness, Orijen, Evo/Innova.

All of the above come in grain-free options and that would be best for hypothyroid and weight loss, and for skin issues.

Feed at least 1/2 to 2/3 of the diet as canned food and the remainder as dry.