They put those Kubotas in Carrier refer trailer units, they go 20,000+ hours. Usually never get rebuilt, get a new trailer at that point.
Rmllarue asked about smoke, is there black smoke? or at least visible smoke when the engine is warm and surging? If there is, then the engine is getting fuel, if there is not visible smoke, the engine runs when it gets fuel and slows when it's not getting fuel.
That's a simple and reliable inline injection pump, I don't know of anything that would cause this inside the pump. It's a mechanical governor as far as I know, so highly unlikely to break in a way that acts up when warm. An injector that's leaking back combustion into the return could do this, but that's unlikely in my guesstimation.
You know that pumping the priming bulb will keep it running, you could try an electric pump to see what that does, but you already know that the priming bulb will do that. It almost has to be a suction air leak, or a weak transfer pump, or a blockage somewhere. Rubber hoses will often work cold and then leak when warm. Maybe an o ring is doing the same thing?
Before even thinking of touching the injection pump or injectors, put a pressure gauge on the fuel inlet to the injection pump, or close to there, and see what the pressure does from cold to warm.