I'm a bit surprised by your measurement of 375A - even momentary - to start a small genset. I wonder if is it an artifact of the sampling rate of the clamp ammeter. A Cummins 5.9L diesel draws only about 300A cranking, motorcycle engines in the 1000 - 1200cc range draw around 150A at stall and 100A cranking. These measured with a 100 KHz amp probe on a 1Ghz recording O'scope. Even if it peaks at 375A for a few milliseconds, that would not heat a thermal fuse enough to blow it.
You might try shutting the fuel off or removing the spark wire from the genset, crank for several seconds and see what the starter really draws. 375A @ 12V = about 6 HP!