My dog stinks like poop and we don't know why.

Hello. We have been having a smell issue with our dog for months now where he just reeks mostly of poop and something else that we can't identify. A couple trips to our vet hasn't resolved this so we are hoping for some ideas here.

Some relevant history. He is a 3 year old black lab, pit bull, great pyranese mix. He is on levothyroxine for a chronically low thyroid hormone levels. He had a skin infection and some ear infections prior to the levothyroxine (how we figured it out). Variations of the bad smell started around then. He hasn't been sick since we got his thyroid in check. At his most recent visit, the vet noticed a couple of patches of dry skin between his back legs, but he otherwise looked good all around. They expressed his anal glands, but that didn't change anything.

His diet is 4 cups per day of Purina one pro plan large breed chicken flavor dog food. Since he is a bit of priss with his food, he gets about 1/2-3/4 cup of shredded baked chicken breast on his food. I grind up his levothyroxine as well as an anti-itch/dry skin tablet (omega-3) and put it on his food.

Any ideas would be greatly appreciated. Please let me know if I missed any key details. Thank you!