I know that this is a TrueNAS forum, but by far the easiest way I have found to flash LSI HBAs is using LSI Storage Authority on Windows.
If you don’t have a Windows machine handy that can hold the HBA, you can use a WinPE boot disk (like Sergei Strelec’s), then install LSI Storage Authority into the running Windows session.
LSI Storage Authority happily takes a BIOS downloaded from the Broadcom site and flashes it to any HBA with the same controller chip with no need to jump through hoops. I just did this to a IBM branded 9300-8i.