Just in the remote chance anyone else hits this issue, the cause was Redgate's Hyperbac monitor.

It had a file association with .ZIP extension files (amongst others). Hyperbac's monitoring of .ZIP files appears to have clashed with SSIS's attempt to write to the .ZIP file, as procmon reported shared file access violations, causing a spurious ACCESS DENIED error to be reported by the package, failing it.

Since use of the tool is necessary on our environments, I was able to solve the problem by deleting the .ZIP association using the GUI ("Hyperbac Configuration Manager" > "Extensions" > Ext=.ZIP, Delete)