This shows how you can use javascript to fetch the assigned group from the case tab and based on the value show an additional custom field on the customer theme.

customer_theme.html

<div class="agent_customer_section" id="my_custom_field" style="display:none;">
  {{custom_my_custom_field}}
</div>

<script>
  $(function() {
    var case_assigned_group = $('#edit_customer_{{customer.id}}')
                                      .closest('#case_sidebar')
                                      .find('#ticket_group_id option:selected')
                                      .text();

    if (case_assigned_group == 'Supervisor') {
      $('#edit_customer_{{customer.id}}').find('#my_custom_field').show();
    }
  });
</script>