Yes low voltage will cause starting issues. Heat pumps use the same compressor as the Air Condition. Do you have a heating strip also. If so low voltage would cause a lot of issues. It seems you do not understand how your Air Condition/heat pump works.
Simply put, a heat pump is a device that uses a small amount of energy to move heat from one location to another. Not too difficult, right? Heat pumps are typically used to pull heat out of the air to heat a RV, home or office building, but they can be reversed to cool a building. In a way, if you know how an air conditioner works, then you already know a lot about how a heat pump works. This is because heat pumps and air conditioners operate in a very similar way.
https://www.gonewiththewynns.com/rv-heating-cooling