If my heat pump does not have a neutral, do I still need a neutral wire?
Neutral is not ground

That thing you are calling a "neutral bar" is bolted directly to the steel of the switch box.  It cannot be a neutral, it must be a ground.

Neutral is an active, live conductor that carries current under normal conditions.  It is normal for it to have "voltage drop", or more accurately "voltage rise".  Ground is a safety shield, and should never flow current except during fault conditions.  

In your installation, your appliance does not use neutral.  Cap the wire off (Put a wire-nut on it) and wrap the nut with tape because they love to fall off single wires.  The stud you have neutral on, is for ground.  A neutral bar would be insulated from chassis, since we are anywhere but the main panel.  

Conflating/interchanging neutral and ground defeats the purpose of running grounds.   People tend to get misled by a couple of things. 


seeing the inside of a main panel where neutrals and grounds gaggle on the same bar.  That is an exception because the main panel is the one location where neutral is bonded to ground to assure the entire system's voltages doesn't float wildly and stays within 120V of earth (which puts neutral quite near earth). 
dealing with 3-prong dryer and range connections where the neutral is attached to dryer frame. That is actually bootlegging ground, and is dangerous, except that it is legalized in those particular installations on the logic that the connections are so rarely disturbed.