I really only have a question about how Cloudflare might behave if I install a 2nd WP site within a new folder under the Public_HTML folder. We are using the Free Plan with Cloudflare.
I have such setups. No issues at all. Only to cross-check and configure per need the Cloudflare Rules if you’ve got some for a particular case or specific paths which has to apply, but which shouldn’t for your /wordpress/ directory.
Are you running WordPress in public_html/, therefrom wanting to have 2nd WordPress installation in public_html/my2ndwp/? I’ve got 4-5 like that, no issue at all, check within the .htaccess file just in case at your hosting.