I'm sorry to hear of this with Conan. There are a number of considerations in such a dog...
1) His teeth might need cleaning; the accumulated tartar is laden with malodorous bacteria which he transfers to his coat when licking himself. Other oral diseases need to be considered as well - stomatitis, e.g.
2) His anal glands might be exuding too much fluid which Conan is licking and then transferring that malodorous fluid to his coat.
3) His skin may be infected. His vet can check a cytology (microscopic exam of Conan's skin surface) looking for abnormal numbers of either bacteria or yeast ande then treat appropriately.
4) Conan's skin may be seborrheic (important differential diagnosis). Here's a thorough synopsis of canine seborrhea for you; not all will apply. There are different types of seborrhea and they can present in different degrees of severity.
Clinical signs may include a dull, dry, lusterless hair coat, excessive scaling (dandruff), follicular casts, scaly and crusty seborrheic patches and plaques, and greasy, malodorous ("dirty gym sock" odor is often reported) skin. Most of the body is involved to some degree, with interdigital areas, perineum, face, axillae, ventral neck, abdomen, and skin folds usually most severely affected. Pruritis (itchiness) is mild to intense, and ceruminous otitis externa (oily external ear canal inflammation) is common. Secondary skin and ear infections with bacteria and Malassezia (yeast) are often present.
1) Ensure good nutrition. A commercially balanced dog food that meets AAFCO requirements should be fed. You should find the AAFCO statement on the food label. I don't believe that his prescription food should be incriminated in his hair odor.
2) Any secondary bacterial and Malassezia skin and ear infection should be treated with appropriate topical and systemic therapies. Periodic treatments or long-term, low-dose maintenance therapy may be needed because these dogs are susceptible to recurring infection.
3) For symptomatic control of ceruminous otitis, long-term maintenance ear care is necessary. Ear treatments with a multimodal therapy (consult with Conana's vet) or ear cleaner should be administered to both ears every 1-7 days to control cerumen (wax) accumulation.
4) For symptomatic control of seborrhea, antiseborrheic shampoos and emollients may be used every 2-7 days until the skin condition is improved (~2-3 weeks), then, bathing frequency should be decreased to every 1-2 weeks or as needed for maintenance. Antiseborrheic shampoos contain some combination of sulfur, salicyclic acid, tar, benzoyl peroxide, and phytosphingosine.
5) Daily oral fatty acid supplementation may be helpful as an adjunct therapy (180mg EPA/10lbs). EPA is thought to be the most antiinflammatory of the essential omega-3 fatty acids. It's plentiful in fish oil supplements.
6) Vitamin A 8000-10,000 IU per 20lbs orally administered with a fatty meal every 24 hours. Improvement should be seen within 4-6 weeks.
7) For dogs with severe, greasy, malodorous, pruritic seborrhea, treatment with systemic corticosteroids may be helpful. Acitretin (a retinoid) may be helpful in some dogs. Calcitriol (vitamin D) may be helpful in some cases.
The prognosis is variable, depending on the severity of the seborrhea. This is an incurable condition that requires lifelong therapy for control.
Playing devil's advocate, I must tell you that crystals are normally found in urine and that the pH of urine can vary considerably throughout a 24 hour period.
Please respond with further questions or concerns if you wish.