At age 13, I should think it was simply important that the doggie should eat something it enjoyed and could cope with rather than some special diet it either didn't like or which made it ill.
When one of mine was 14 and his kidneys started to pack up and he lost weight, the vet put him on a Hill's special diet - which he hated and refused to eat. So the very sensible, hands-on elderly vet told me, just feed him whatever he will eat, at his age. Make him happy.
So I put him back on the same diet as all the rest of the pack - he went back to eating, and even regained some of the lost weight before, in the end, events caught up with us and he was pts.