FF6Tools - Browser-Based ROM Editor

Checksum can be changed, I think what counts is that the checksum has a corresponding inverted one, meaning you can use whatever set of the four. You'll likely never have the good checksum with a patch that change it or after a manual edit (in this case it remains the same but becomes no good), but some utilities like FF6Tools likely recalculate the good one and it will be different each time you change a single byte in the ROM.