I’m not quite clear on what you’re asking for regarding the ASM who signed off on the rank. Generally, we have signatures on the scouts’ handbooks from various ASMs, the Scoutmaster, and senior scouts who have been authorized by the ASM to sign-off on things (e.g. SPL, ASPLs…). We typically have the BoR chair sign the handbook at the completion of the BoR, since it is the Board that is approving completion, rather than the SM/ASM in that context. I have my scouts enter the name of the person who signed-off their handbook in the Notes field on Scoutbook when they mark something as Complete, so we can backcheck if needed.
Scoutbook will record who “Leader Approved” any given requirement on Scoutbook. Are you looking to have the proposed report display that Leader Approved name?