@DelightedHippo I’d never been able to get satisfactory results with any of the static site generators or plug-ins or Gatsby front-ends with WordPress until I tried using Frontity and Vercel. Your mileage may vary.
Yes, you will still need to hold your WordPress instance on WPEngine. Depend on your billing tier, you may save a few bucks because your traffic / visits will go down, but if you have 25 sites there, you’re still paying at least for that basic tier. The biggest change you will have to make is to revert the URL of your WordPress install to wordpresssite.wpengine.com (the default “internal” URL for your WPEngine site), and then point your custom domain to wherever you host the static version.
The best way to do this, IMHO, is for the static site code to pull from your WordPress JSON file, which is only updated when you make changes to your WordPress site. Note that if you use Disqus for your comments, you’ll have to account for that. The trick is getting your static site generator to duplicate the look and feel of your WordPress site, but with faster-loading pages. This can be a huge hurdle, especially if you are not a programmer.
If you don’t have the skills to make Gatsby or Frontity or whatever dance for you, and you don’t feel like paying a developer a fair chunk of change, stick with WPEngine.
If you’re looking for more site speed for cheap for your WordPress site(s), Cloudflare just introduced a new CDN service that is better than WP Rocket (and etc.). If you are already on a paid Cloudflare tier, it’s free. If not, it’s five bucks a month per site, and you won’t have as many caching issues as you do when you are running WPEngine cache and Sucuri cache and WP Rocket cache.
My two cents.