support might be able to accomplish this for you... but I'm guessing they can't. You're probably stuck making backups and everything, deleting the domain from your DH account, and then 'restoring' to your new domain.
Just a side note, do you know that you can add a "booster pack" to your plan for only $60 a year or 4.95 a month (w/ a &15.95 setup fee) that will allow you three more domains and some other stuff.
I was just thinking (suprising stuff!) that you could just set up a mirrored domain to point to your existing one. That way you can have all 4 domains with in your plan - the new domain will allways reflect your current domain name.
ie. you've got bobby.com set with with your software and all. You've just registered dobby.net - so you set it up as a mirroed domain for bobby.com and it will give you two identical sites. You just make changes to and edit bobby.com and dobby.net will access it's stuff, while showing your visitors dobby.net in their url.
Maybe this will work out for you.