The batch number should not die at spawning. It belongs on the room sheet, the spawn lot and the yield record, because that is the only way a third flush problem traces back to a formulation change six weeks earlier.
Paper is fine for a small unit if it is complete and legible and kept where the next shift can read it. What matters is that the record was written while the work happened. A record filled in from memory on Sunday evening is a story, and stories cannot be investigated.