Or instead of doing steps 2 and 3 above, you could just use a redirect rule to your advantage here to swap out your old landing page with no downtime.
Just make your new landing page on the new template, then create a new rule that redirects www.landingpage1.com to www.landingpage2.com; this will take effect immediately and you won't have had to manually swap out the URLs on either page. Just make sure to also update your triggers to react to the new landing page.