There is no fuel pressure switch in there at all. The only sensor in the fuel system is a fuel temperature sensor on the fuel rail. It's a recirculating type system with a bypass regulator, normally it pressurizes the system and then shuts off after a fixed amount of time if you didn't start the engine. But it runs constantly while the engine is running, any extra fuel pressure is released by the regulator and run back to the fuel tank via the return line. So you're not going to burn out the pump, it's designed for constant use.
I'd start with testing the fuel system relays next to the ECU. There are two silver ones next to each other, one controls the O2 sensors power and the other the fuel pump. Swap them around and see if the problem changes.
My oil line fix
My 100k stopped odometer fix
Brace for impact, I'm in a "modifying mood". Massive thread count inbound.