I don't know where to start. I quite know the answer but just cannot put them into words. Need help...
I think #3 nails it on the head in pointing towards what Miss Emily represents symbolically in this story. She represents the fading values and culture of the deep South - a living anachronism that is soon to die out. Her obsession with Homer Barron that leads her to kill him has nothing to do with the fact that she is fat - rather her fatness represents her fading out of this world. Note that she only begins to put on weight at the same time that her hair turns grey after she has killed Barron.
Let us start by rephrasing the question somewhat. "Why" sounds like we're asking about the author's intent, or about the creative process. That may be lost to us. Instead, ask "What is the benefit of examining obsession in such a character?" Then we get answers.
The benefit is, she is the exact opposite of someone who we might think of as capable of obsession. She's a little old lady, for heaven's sake! She's from a good family, they've been around forever, etc. She has no external reason for obsession. That means we have to look deeper, and look inward, and that the final answers are more striking.
Faulkner had Miss Emily represent the fall of the old south. The fact that Miss Emily is "a fat, strict, old woman," has nothing to do with her obsessions, if she truly is obsessed. The age, weight, or flexibility of a character or person has little to do with their obsessions. If you re-read the story closely she was rather young, thin and shy when the story begins. Her father is strict, and her life is one of a "conservative southern lady." Time moves on for the south and values begin to change. Miss Emily can't let go of the past. She tries to hold on to her father by not letting the town's people bury him, but eventually realizes that she must. She kills her lover. Did she do this because she was obsessed, or did she do it because he was going to leave her and she couldn't bare to lose another person in her life. Miss Emily simply can't get away from her past. She becomes, old, fat, and strict because of her "obsessions," not the other way around.
Perhaps Miss Emily's being fat does have something to do with obsession. Afterall, people can have food obsessions, yes? But, here in the context of this story, Emily's heaviness may indicate her sedentariness in the mores of the Old South, her figurative remaining solely in the one position of daughter of Colonel Sartoris and figurative nurturing on a dead culture that offers little nutrition; this obsessive feeding on the past only clings to her frame as useless fringe, represented as fat on her old body.