I am confused about what you’re talking about when you say that CPanel is incompatible with Dreamhost. You are correct in that they don’t use CPanel here, but I’ve never seen any features of CPanel that would let you specify a transfer destination.
Perhaps you are talking about the files that you get when doing a full backup from CPanel. If so, there would be no place to import that huge backup monstrosity that CPanel produces. Plus a full restore wouldn’t work anyway because the paths would be wrong.
Your best bet is probably to go into a shell and archive up the files that you want.
Which is essentially what Ryo-ohki said I guess. You’re likely to have the best luck grabbing exactly what you’re looking for instead of trying to transfer everything. Perhaps this wiki entry will help. If not maybe someone can offer more suggestions with more detail from you.