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OK sorry, this was quite simple finally.

In case this can help someone spare some time, all I did was :

public function hookDisplayProductAdditionalInfo($params)

{
    ...

    $html = '';

    foreach($available_extra_fields as $available_extra_field) {

        if($available_extra_field['location'] == 'additional_info') {

            $html .= $this->displayProductExtraFieldContent($id_lang, $available_extra_field, $product_extra_fields);

        }

    }

    return $html;

}