A little bit of further testing would seem to indicate that it matters when AA is toggled on in relation to the edit session...
I first edited the config file to alter the following, in order to maintain explicit control over the order of operations:
"AttributeAssistant_EnabledOnStartUp" value="False"
"AttributeAssistant_EnabledOnStartEditing" value="False"
"AttributeAssistant_ClearLastValue" value="True"
"AttributeAssistant_ClearLastValueStopEdit" value="True"
I reopened my edit MXD, making sure that AA was off. I started an edit session, and then toggled AA on. At that point, the CURRENT_USER method will grab the appropriate database username. If this is done in reverse, or if AA is already toggled on, prior to an edit session being started, it seems that the CURRENT_USER variable remains unset/null.