Hi finally, I could do it. I went back to the simple solution and added
<div>{{ this.order.profile.membership.name }}</div> to the skins/common/order/invoice/parts/head.twig and it displays. i don't know why did did not work in the note.twig. Thank you everyone for their awesome support.
{##
# Invoice head
#
# @ListChild (list="invoice.base", weight="10")
#}
<table cellspacing="0" class="header">
<tr>
{{ widget_list('invoice.head') }}
</tr>
<tr>
<div>{{ this.order.profile.membership.name }}</div>
</tr>
</table>
Kind Regards