I answered my own questions. You need to use:

$bulkCopy.ColumnMappings.Add("SourceCol", "DestCol")

to define the mapping between columns.

I changed the script to pull the server name from __Server instead of SystemName because a few of my servers had no information in that field. After doing that...

I just found this post and I am trying to set everything up using the lowest privileges. But I keep getting an access denied error. Does the user...