I partly agree with Stevie D, regarding the nonsense post that a blanket statement that WP will make it better simply because of the magic combination of PHP/WP however in many cases you can leverage the CMS and make a marked SEO improvements with more flexibility in page titles, navigation, meta tags (if you still believe in them), XML-Sitemaps, keyword rich URLs, etc... I have some experience in this area and have made really poor performers turn around and become top performers. However, as you know it's more difficult when the website is already a top performer.
My chief concern would be how many pages does the site contain and how many of these pages are the ones that perform well in the search results. It's likely that only a fraction of the pages are performing well in the SERPs and those are the ones to be aware of.
One part of my strategy would be to use the .htaccess to do a 301 redirect from the existing paths of top performing pages to the new ones.
RewriteRule ^old-page.htm http://www.example.com/new-wp-url [R=301,L]
The rest depends on why the site gets good SERPs. I'd spend some time figuring that out and then makes sure that the new site retains or improves on those areas.
Note: Don't forget just because a site does well this week or the last 6 months doesn't mean it will last. Google does updates and people move around in the ranks. I used to have my business website in the #1 spot for the terms I chose but over the past year I've gone anywhere from 4th to 24th and currently I'm #14, so on page 2. It's fine because my site is well over 10 years old and needs to be replaced but it's something to keep in mind... When you're #1, you can only retain or move downward.