Step 1) Take A Full cPanel Backup Before Getting Started cPanel -> Backup -> Download Full Website Backup
Step 2) Install a NEW copy of WordPress inside a subfolder (i.e. /wp/) using Softaculous cPanel -> WordPress (under Softaculous) -> Install
Step 3) Locate your CURRENT WordPress installation's location and open up the wp-config.php file inside of cPanel -> File Manager
Step 4) Go back to cPanel -> WordPress (under Softaculous) and click on My Apps
Step 5) Click the pencil icon beside the WordPress you just installed
Step 6) Modify the details to match your original WordPress installation (in my case removing the /wp/) and using the database information from the original wp-config.php file.
Step 7) Enable Auto Upgrade, Auto Upgrade WordPress Plugins, and Auto Upgrade WordPress Themes
Step 8) Save Installation Details - Congratulations your WordPress website will now automatically upgrade using Softaculous!
Step 9) Remove The Database and Database User of the "other WordPress installation " cPanel -> mySQL -> Delete
Step 10) Remove The Folder of the "other WordPress installation" cPanel -> File Manager -> public_html -> wp (click and select delete):
You can now take full advantage of other Softaculous features such as one-click cloning!
Manually Add WordPress To Sofaculous Video Tutorial
I filmed a video tutorial with all of these steps so you can follow along: