The inability to access package variables renders Package Parts pretty useless, actually
I haven't used SSIS for a while but I remember similar behaviour from elder versions (SQL Server 2003). It wasn't possible to assign values to package variables at runtime directly. However it has always been possible to assign values via a script task. I'm convinced that a script task can pass package variables to exposed package part variables.
You should give that a try. If it was possible 10+ year ago, why should it not be possible nowadays?