Hi,
really you can download the config via snmp? are you sure? the MIB: CISCO-CONFIG-COPY-MIB.oid is normally the way how to initiate the download of the running-config. I normally set the specific option just mentioned it here: http://ccie20728.wordpress.com/2008/05/20/get-the-cisco-configuration-over-snmp/ But I never configured it with an ASA - only with Cisco Switches and Routers.
Unfortunately in MIB Support List from Cisco for their ASA this MIB is not listed: ftp://ftp.cisco.com/pub/mibs/supportlists/asa/asa-supportlist.html
Also the Nexus Switches have not implemented this Mib until yet.
So I am sorry but I believe you downloaded your config via SSH and you have an issue with your connection profile or with a limitation at your ASA.
Kind Regards,
Flo