If Outlook's adding the header, you could always switch to something better. Even something free like Thunderbird.
Otherwise, if there are just a few emails you'd rather send without that header, just log into Gmail & send those... since you already setup the other send-from identities there. Then you don't really have to change what you're doing now.
If the main goal is to take Dreamhost's mail out of the picture, then I'd use Google Apps (or any 3rd party mail host) and just do everything through there, rather than using forwarders & leaving stuff in an inbox here as well.
Maximum savings promo code: MaxSavingsAtDH