Well to be exact the package is failing on the last fact table I'm trying to fill, it's data is mostly based on the previously finished fact table.
Well the source for the other tables begins with either external DB or Flatfile, goes through transformation into dimension tables and from there on to the facts.
In this case for the previous table the source are dimension tables within the same DB, and both (Dimension and Fact tables) are within the same dbo schema, we only use other schemas for extract and transform. There is about 15 dimension tables being built before any of the Fact Tables are being built.
Meanwhile I can confirm that adding the SQL Agent Service account (to sysadmin) does make the message about permissions disappear when running the statement with EXECUTE AS LOGIN= but the last attempt via execute SQL Task didn't yield any inserted rows unfortunately.
The latter to me makes sense because otherwise there would be no reason why the other tables are treated differently (rows are inserted correctly). Even if the SQL Agent Account would come as NT SERVICE\SQLSERVERAGENT, this account has per default sysadmin permissions.