Try logging progress to both default output (STDOUT) and global filehandle LOG with LOGress :)

BEGIN {
    my $started = !!0;
    sub LOGress {
        my $string = join '', @_;
        my $before = tell select;
        print $string;
        my $after = tell select;
        if( $started ){
            seek LOG -abs( $after - $before ), 2; # rewind
        }
        print LOG $string;
        if( 0 <= index $string, "\n" ){
            $started = !!0;
        } else {
            $started = !!1;
        }
    }
}
...
LOGress "\rProgress...$x\%";
...
LOGress "\rProgress...Done!\n"; # \n signals end
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use autodie qw/ open close seek tell /; to die if those calls fail