I would read them in that order. The endings are nice and complimentary too. I think McCarthy read Lebowitz and took some inspiration. Lebowitz is also a good palette cleanser after the tender bleakness of the Road. Intellectually stimulating and different enough while in my mind being a non cannon sequel to the Road. Love both books so much. Good little post apocalypse pairing
By coincidence, I just picked up Canticle for Lebowitz in a local bookshop. Your post feels like intuition confirmed. Unfortunately, the ‘read it at the end of the sequence’ proposal falls. I may try The Road afterwards and compare notes.