If you have small syslogs, it will fill more quickly. I assume your doing a bcp out and bcp in.
What you can do, if not done already, is configure this option "trunc log on chkpt" in order to truncate the syslogs everytime a chekpoint occurs. ( 50 lines inserted or once per minute). This way you should have the syslogs with low usage.
In my opinion you should also remove your indexes to insert the rows. After the insertion recriate them.
This way you get a fast bcp in. Otherwise with indexes you won't.
About that named cache, it won't help you on inserts just on work on that db, queries, temporary tables, indexes creation...
Did you bind that named cache ?
sp_helpcache - check cache binding information.
Hope it helps