Support

 I am also concerned that by changing the PHP versions in SPanel it may overwrite any security that Admin Tools provides.

Let's start with putting your mind at ease. Changing your PHP version in cPanel has no impact to your security. cPanel's multi-PHP versions feature is actually fairly simple. First, it looks to see if it's already modified the file (it puts a comment line with a specific structure). If it's already modified it, it replaces its own lines only with the new ones. If not, it appends its own thing at the bottom of the file.

The only thing cPanel adds / modifies is an AddHandler line which tells Apache which predefined FastCGI executable to use to parse .php files. It just wraps it inside an IfModule block because cPanel is designed to be, um, idiot-resistant. Some people who fancy themselves as hosts shouldn't be allowed near a computer, but that's another story and I digress.

Would it be easier/better to re-run the Quick Setup Wizard and import the configuration settings?

No. The Quick Setup Wizard resets everything to factory defaults. You'd have to redo your site's security afresh. Don't punish yourself like that.

I would follow a different process.

1. Set the new PHP version. This creates a new .htaccess file with the requisite AddHandler lines.
2. Open the .htaccess file and note down the AddHandler line.
3. Create a new database and database user. Remember to connect the DB user with the database, allowing all privileges cPanel lets you assign.
4. Upload Kickstart and the backup archive. Obviously ;)
5. Use Kickstart to extract the backup archive, then go through the restoration process. During the restoration choose to reset the .htaccess file to the Joomla default. Note that Kickstart and ANGIE are aware of the AddHandler line added in step 1 and carry it over to the new file. Therefore, your site will be running under the correct PHP version (server willing...).
6. Log into the restored site's administrator, go to .htaccess Maker and paste the line from step 2 into the code to add at the bottom of the file. If you had a different AddHandler line there, replace it.
7. Regenerate the .htaccess file.

At this point you have the security-enhanced .htaccess file created by Admin Tools, with the AddHandler line to apply the new PHP version on your site.