> the varroa mite feeds not on the bee’s haemolymph but on their “fat” bodies.

I am not sure why people think this has to be one or the other. Normally, when you eat an egg you eat both the white and the yolk. Fat bodies are in the hemolymph, so if the varroa is sucking on a bee pupa, she's getting a mouth full of hemolymph and fat. 

> The insect fat body is a mesodermal tissue composed of a meshwork of loose lobes suspended in the hemocoel and bathed in the insect hemolymph. The tissue is composed primarily of vacuolated rounded or polyhedral cells called adipocytes or trophocytes, which commonly harbor stored inclusions of proteins, lipids, and glycogen. 

> It is noteworthy that the fat body is intimately associated with nearly all vital tissues and organs in the insect body, including the tracheal system, the musculature, the Malpighian tubules, and the hemolymph. This spatial organization is well adapted to the physiology and the open circulatory diffusion system of insects, thereby facilitating absorption and release of metabolites and nutrients.

>  During the prepupal period, mass quantities of reserve material are accumulated in the fat body cells. Lysis of fat body cells in higher dipteran species at metamorphosis results in the discharge of stored reserves into the hemolymph. However, as the new adult fat body cells are formed, nutrients are reabsorbed.

> Dynamic exchanges of nutrients between fat body cells and the hemolymph compartments are evident throughout the life cycle of holometabolous insects (= their life cycle includes four stages: egg, larva, pupa, and adult.)

PLB

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