ok, I figured it out myself.
instead of using the CSS class (as I cannot figure out how to filter it), I decided to use the description instead.
under the lines with $attributes I have added this line:
$description = ! empty( $item->description ) ? esc_attr( $item->description ) : '';
And then I added that to the output as:
$item_output .= '<a'. $attributes .'><i class="icon-' . $description . '"></i>';
I would still be interested to hear whether it can be done with class too?