The Cycle Sensor is usually telling the truth. It is watching 27/7/365 and is seeing something you are not. Maybe the pressure switch it too far from the tank or too much air in the tank is causing the switch to bounce on startup. Setting at 60 seconds won't make any difference if it is a bounce. But if it doesn't trip on rapid cycle at 60 seconds, I would move it up to 110 seconds and try again. If you know the tank takes 2 min 40 seconds to fill when no other water is being used, then it should take at least 2 min 40 seconds every time. Sometimes the little problems are the hardest to find, but they need to be found.
A 1HP, 5 GPM pump can build 290 PSI. So pipe would need to be rated for that to work with a CSV or even just to set the pump at 500-600 feet deep.