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.

It sounds to me like like you're about to violate some NEC/NFPA* fire and safety regulations. Replacing a fuse like this with bare copper wire is ILLEGAL period. Machine downtime is always preferable to property damage, injury or loss of life.

EDIT * Assuming you're in the USA.

Forget trying to design your own fuse, leave that to experts whose livelihood depends on safe and accurate designs. Call a fuse manufacturer's technical department and discuss the parameters that you provided on your load (600A 5 secs maximum, 50A continuous).
Fuses protect the load AND the cable feeding the load. Replace your parallel fuse holders with one fuse that is designed for purpose.