It looks like you have a pretty good handle on the issues, and certainly one way to approach would be the "wipe" and redo, though I understand why you would rather not go that route (having done it myself more than once!)
One of the "quirks" of the "one-click" installs is that, in order for the Dreamhost "auto-install" scripts to handle them, and their updates, they do require their "target" directories to be empty. This is really only necessary when installing these applications using the "one-click" installer; if you install them traditionally (they are very automated already ), you can put them in a directory that has other directories and files (you just have to make sure you don't have more than one file with the same name . What this generally mean in practical use is that one application lives in the "domain.tld" directory, and the other live in either sub-directories or sub-domains (because of conflicting .htaccess files, re-write rules, index.php files, etc).
Since you have already built the required database (as part of the "one-click" install), you might consider just installing the code yourself by grabbing the latest Joomla package from joomla.org. Or, you can use the panel to "create another database", and then proceed if you wish.
While installing Joomla! yourself might seem to be complicated, it is really very simple and if you learn to do that, installing your own instances of Joomla!, WordPress, and other scripts yourself will give you a lot more flexibility.
I suggest you take a few minutes over at joomla.org reviewing the installation instructions before you decide how to proceed. You may well decide to "give it a go" yourself ion order to get the directory structure you want without a complete rebuild of the domain. Good luck, and post back if you want need more help or discussion.
Using sub-domains is also an option you should not overlook, as that gives you a lot of possibilities (Joomla on domain.tld, forum.domain.tld for your forum, blog.domain.tld for your WordPress blog, etc.)
I also think that, at the end of the day, you will not really be happy with "forcing" the url to be what you want via a "hard" redirection - it will add a small amount of additional latency to your site's response, and could, down the road, complicate things considerably as that redirection interacts with the re-write rules many of these applications (wordpress, joomla, etc) use to "prettify" their urls. While it may seem to be extra work to get it sorted now, it won't be any easier after you have all the applications successfully installed. If it were me, I'd bite the bullet and get it set up the way I really want it even if I do have to "wipe" and "reload" - but that is just my opinion, and YMMV!