Given those details, I could see the Avatar wanting to block its senses (put out its eyes, destroy the nose) so as to not easily perceive food.
That said, I wouldn't really consider the cursed person to be an avatar of Gluttony, as I generally consider an avatar to be a literal embodiment of something (here, a concept of gluttony.) That said, I could see trying to loophole around it by some of the things others have suggested, in mentally altering one's definition of food so as to exclude certain things, or try to shift towards gluttonous concepts like being a gourmand/foodie (caring about the particulars of a food more than its quantity).
Through meditation or focus, one could probably get used to unending hunger. Someone with the Mantle of Gluttony being a blind, scarred monk without a tongue, who lives on bland paste put in his mouth at the minimum to stay alive... that seems a doable concept with this idea.
EDIT: to add to your first question: I don't think you could go on a diet, as you don't have the ability to not eat food that you see. Unless you can change the definition of 'food' with respect to how the curse operates.