What fuses can I use for branch circuit protection above 600 amps?
I'd consider using a circuit breaker rated for branch circuits but you can also get fuses. See this from Cooper industries: -


  Standard sizes for fuses and fixed trip circuit breakers, per 240.6,
  are 15, 20, 25, 30, 35, 40, 45, 50, 60, 70, 80, 90, 100, 110, 125,
  150, 175, 200, 225, 250, 300, 350, 400, 450, 500, 600, 700, 800, 1000,
  1200, 1600, 2000, 2500, 3000, 4000 5000, and 6000 amps. Additional
  standard fuse sizes are 1, 3, 6, 10, and 601 amps.


This one goes up to 700A: -



From this page.