what size of copper wire can act as a 150A fuse?
Finding a fuse with those characteristics is going to be difficult. I'd use a current shunt with a suitable MCU and circuit breaker and monitor the current taken by the device, shutting off the current if it rises above 120A for 15 seconds.

Two fuses in parallel can cause problems unless they are well-matched and the holders are properly designed. Any difference between them can cause one fuse to take more current than the other one, and fail.