Problem solved.
I logged into WHM on my Centos64bit VPS as root, and from withing WHM I ran the EasyApache process, and in the advanced settings there was an option to recompile Apache (and PHP and lots more).
In there I found a checkbox for php_mysqli or something…
After it ran the recompile (about 3 minutes), and nothing was broken, the BackWPup now works GREAT.