You should simply be able to pass $file_list to du:
du -ch $file_list | tail -1 | cut -f 1
du options:
-cdisplay a total-hhuman readable (i.e. 17M)
du will print an entry for each file, followed by the total (with -c), so we use tail -1 to trim to only the last line and cut -f 1 to trim that line to only the first column.