Hm. A few ideas...
Why don't you want to use mirror? It's the quickest way...
You only need to go to the DH Web Panel > Manage Domains > Add New Domain, then scroll to the bottom and use the Mirrored option: place site2.com as your "new" domain and then select site1.com from the dropdown. This will actually get you both domains pointing to the same place and use both addresses.
Then you will need to fix your links. You didn't say if you're using WordPress for your site or if it's just a static website. WordPress is not so easy to "automatically" change the domain name. Here is a list of the procedures to do that:
or apparently there is a script for that, too: http://dan.doezema.com/2010/04/wordpress-domain-change/
If you're using a static website, of course you will need to change everything manually, unless you have been clever enough to do all links relative
I haven't used FileZilla for a long, long time, but I'm pretty sure that what FileZilla will do is what @Saldash said. It will copy everything from site1.com to your computer first, and then you upload everything back to DreamHost into the new location (for site2.com).
If you want to avoid this, and if you're familiar with the Unix command-line shell, you can do everything on DreamHost's side — it will be much faster. You can log in to site2.com, change to the site2.com directory, and then use
rsync -av email@example.com:site1.com/* .
to copy everything from one place to the other; "username" is the username you have for FTP for site1.com. Remember, you will have to go to the DH Web Panel and enable SSH for "username" for this to work.
If you're not comfortable with the Unix command-line shell then your best option is doing exactly what @Saldash said... it will take longer, but you'll get a copy of the whole site on your disk as a bonus