Since the temperature of the organism is essentially maintained while breathing, is the work done by the organism considered isothermal?
The process is not isothermal, the air heats up when incoming and outgoing. Nor it is adiabatic. I would say in a pV diagram it occurs at (almost) constant pressure (when the diaphragm moves, almost concomitant air enters the lungs, filling it, so maintaining a certain equillibrium between internal and external pressure).
The temperature of the body is maintained the same by the liberation of internal energy of the ingested carbon chain molecules in the inner celular structure, what compensates this and other heat changes with the environment (such as radiation).
But it takes care also to assure that 'the temperature of the body is always the same', as long as, there is no single temperature for it. For example the skin tends to be a few degrees lower then the viscera. All in all, there are models that consider the body a kind of 'onion-like' structure, in relation to its temperature, going from inside shells (in which temperature is higher) to outside.
Since our body temperature is fixed at 37degree celcius and breathing is done inside the body so under normal conditions, breathing is isothermal.