You’re putting mayo on a grilled cheese sandwich (ymmv, but gross) and you’re worried about the fat from butter on the bread?
A place near work makes grilled cheese sandwiches with no oil of any kind. they put the cheese (and any other ingredients into a sandwich and then throw the sandwich into their pizza oven (so it is more of a toasted cheese sandwich). They aren’t bad sandwiches, and the place is using really good quality ingredients, but it still is lacking something in the flavor department.