I think that Plesk doesn't allow any site applications to be installed in the webroot httpdocs directory. It will require a subdirectory to be specified for the install.
You would have to copy all files from your current site and upload your files to your new Lunarpages account. LP do provide temporary url. It will come with your welcome email. You can test the account by viewing your site via temporary url. Once the site is working successfully, you can then change your DNS nameservers over to your new account .
Thank you for choosing Lunarpages.