Just create another Gmail account. Link it through POP or IMAP to lift all mail from certain folders which contain big attachments to the the point that it takes about half the load from your main account, and let it do the job overnight while you dream of how cool this is.
For example, I have a Gmail account called "email@example.com". It takes only a few clicks. I use it only to lift all my sent mail (which invariably contains big attachments, but which I MUST also keep as proof, for the off chance someone disagrees with me about the content I sent). In this way, my main email account is not going to get full any time soon. It's not really like you are abusing the system. You should also look into cloud management (I know a guy who has 35 Gmail accounts and centrally manages through an app all the dropboxes of these accounts, making it seem as though he has half a terabyte online).