Have you tested with this wireless external drive not connected?
Yes, makes no difference.
In the other User Account, hold down the Option Key while clicking to launch Aperture. What libraries are offered in the resulting dialogue? Is the primary library that you use in your normal user account listed? If so, can you choose and open it? (there might be a legit issue of permissions with that, but check)
It isn't listed, but I can navigate to it. I can't open it, so I navigated to it in the finder and allowed myself read and write permissions (in fact I allowed the whole file path). But oddly that doesn't work. It says that the permissions are wrong. When I go back to my main user account to check on the permissions, it recognises that my new account has read/write access. Not sure why it's not letting Ap open it in the new account.