Root_Loader.php uses file_exists($filename) before include_once($filename). file_exists() returns true for directories, which can lead to PHP warnings when $filename resolves to a directory or empty value. The minimal fix is to use is_file() and ideally also is_readable() before including.
Patch:
Replace (in /w3-total-cache/Root_Loader.php)
if ( file_exists( $filename ) ) {
with
if ( is_file( $filename ) && is_readable( $filename ) ) {
This prevents directory-includes and avoids PHP warnings. I can provide a small patch file if helpful.