If you eat enough protein, you probably could do precisely what your model suggests. Muscle 'damage' will require amino acids for new protein synthesis, and fat cells will provide free fatty acids which can be used for energy for most cells, but aren't useful in protein synthesis. So the calories from fat cells can't directly provide protein for muscle growth.
But I'd imagine that if the body has a choice between burning free amino acids or using stored fat, it'll go for the amino acids. So the amino acids will be used for energy and won't go to the muscles. At least, the majority of it won't (probably just enough to maintain the muscles, not to stimulate growth).
"Thanks to the combination of meat, calcium-rich leaf foods, and a vigorous life, the early hunter-gatherers were robust, with strong skeletons, jaws, and teeth." - Harold McGee, On Food And Cooking: The Science and Lore of the Kitchen