The formula for the cause of a hot water only odor created in water heaters is as follows:
Sulfate is present in the water. A sulfate reducing bacteria is present in
the water. This bacteria takes the oxygen off of the sulfate, making it
sulfur.
The anode rod generates free hydrogen in water.
The hydrogen and sulfur together produces the smell. It is H2S (hydrogen sulfide) gas described as sulfur, rotten egg, sewer type odor.
Removing the anode rod will remove one part of the equation as long as you don't strip off some of it into the tank when removing it, eliminating the odor. Increasing the temp to 140f for an hour or so will kill the bacteria. Most water heater manufacturers also have a different type of anode rod available that does not generate the hydrogen - you would have to call them and see what is available.
Sanitizing the hot water heater with a heavy chlorine eliminates the
bacteria and produces a temporary relief.
To quote Wes McGowan, excerpt taken from Water Processing for Home, Farm and Business 1988:
When a hydrogen sulfide odor occurs in a treated water (softened or
filtered), when no H2S is detected in the raw water, it usually indicates the
presence of some form of sulfate-reducing bacteria in the system. These
anaerobic, single-cell bacteria (Thisbacilles) can exist in the piping system
in the home, especially on the hot water side. It is most noticeable on the
first hot water drawn in the morning. Sulfate bacteria can derive energy by reducing the sulfate ion in the water to H2S, and produce by-product bicarbonate in the process. Organic matter needs to be present for the bacteria to survive. However, the concentration of organics in the raw water is often below detectable levels. When this condition of H2S in hot water arises, the initial task is to heavily chlorinate the entire piping system including storage and hot water tanks. Usually, a dose of household bleach left standing in the piping system (hot and cold) over night will destroy the sulfur bacteria.