Kiln's actually pretty smart about that: if you do the branch on the server—that is, you choose Branch from the little drop-down menu next to a repository—the new repository should use almost no additional disk space. That's because Kiln and Mercurial use hard links when they're making clones on the same machine. Hard links basically allow the same file on disk to appear at multiple places on your file system. When you do a server-side clone in Kiln, we simply hard-link all of the data in the old repository to the new one.
As you take the two repositories their separate ways, the linking gradually degrades over time, since the parts of the repository you modify will have to be copied as they begin to differ, but you should see nothing anywhere like exponential growth. If you think you are, please let us know so that we can take a look at what's going on.
I am not entirely sure what you mean by "clone from the named branch on my local repository", but you can clone a single branch and all its ancestors using hg clone -r.
Named branches also allow for branch-per-feature development. Based on your description, it sounds like you are attempting a variant of this. We have been using named branches in a branch-per-feature development process for several months. It's working very well for us. Our process is described here: