Dyshidrotic Eczema (Pompholyx): This condition is characterized by small, itchy blisters on the palms of the hands or soles of the feet, but given the description of the rash being not itchy and localized to the palms and extensor surfaces with a specific appearance, it's less likely. However, the mention of working in a gym and the presence of a healing cold sore might suggest a stress-related flare, which can sometimes present atypically.
Palmar Psoriasis: Given the description of the rash being on the palms, with a circular appearance and dusky red centers, this could be a strong consideration. Psoriasis can be triggered by stress, and the patient's recent cold sore could indicate a period of stress or immune system activation.

Secondary Syphilis: This condition can cause a rash on the palms of the hands, among other places. The rash of secondary syphilis is often described as maculopapular and can appear on the palms and soles, which fits the patient's presentation. The recent cold sore could be a coincidental finding or part of a broader immune response.
Gianotti-Crosti Syndrome: This is a rare condition but can be considered given the patient's age and the description of the rash. It's characterized by a papular rash on the face, arms, and legs, often following a viral infection, which could include the herpes simplex virus causing the cold sore.

Syphilis (again, mentioned for emphasis): Due to its potential for severe long-term consequences if left untreated, syphilis must be considered seriously, even if the likelihood seems low based on the initial presentation.
Lyme Disease: Although less common, Lyme disease can cause a rash (erythema migrans) that might initially be mistaken for other conditions. Given the potential severity of untreated Lyme disease, it's crucial to consider it, especially if the patient has been exposed to ticks.

Keratoderma Blennorrhagicum: A rare condition associated with reactive arthritis, characterized by a rash on the palms and soles. It's less likely but could be considered if other symptoms of reactive arthritis are present.
Pityriasis Rubra Pilaris: A rare skin disorder that can cause a variety of skin manifestations, including on the palms. It's characterized by small, pointed bumps and orange-red patches, which might not exactly match the description provided but could be considered in a broad differential.