Jason,
Another workaround is to replace the code of the current landing page template with the code of the new template. So you're basically replacing the current template with a new design. Once you approve this landing page template, all of your landing pages that are based on this template with go into a Draft format.
Review the Draft of the landing page you want to change the design for with the new template, modify any font placement etc, and then approve the draft of the landing page. This way your LP will maintain the same URL.
If you don't care much about maintaining the same URL as the original landing page, it's more risk-free to create a new template with the new design altogether and then choose this as the template for the cloned landing page.
Hope this helps.
Thanks!!
Kashmira