Why did you opt for a snowflake schema? There are limited uses for this in an OLAP design, but most designs are properly down using the star schema.
As for the composite key, you need to eliminate it in favor of a single field generated identity key. You can unique key the two fields if that's a true business key and any updates to the other fields will simply get replace. Once you've done this, you won't have any trouble making the joins in SSAS.