Ok, so only some of the mail gets forwarded. I wonder if crontab bypasses any forwarding, but that's strange. Maybe it's possible to use two addresses in crontab's MAILTO statement at the top of your crontab file:
Or as a bigger experiment, create another DreamHost mailbox and set your primary mailbox to be forward only to your Gmail account and your new DreamHost mailbox.
What other mail doesn't get forwarded? Stuff mailed from outside of DreamHost? Other than moderation notices, what else does get forwarded?