All of this:
The first INSERT statement moves about 800,000 rows from these tables. The second INSERT statement creates about 200,000 rows in the fact table by querying the fact table itself.
Is written to the log file before it is written to the database (not completely true, as some data may get written to the database before it is all written to the transaction log). The reason for this is in case of a failure and the transaction needs to be rolled back before it is completed (such as a server crash).
If the transaction log has grown to 9 GB during this process, then that is how large it needs to be. If you are concerned about the size of the transaction log, can this stored procedure be modified to complete the load in smaller batches, say of 100,000 rows each instead of 800,000 and 200,000 respectively?