[cflsystems]

You need to create custom module for this, best option, and add new template file to the invoice. You can position it by using @ListChild in the doc block at the beginning of the file. Just see how other templates are done.

This is a quick dirty hack. Using customer/order/invoice/parts/note.twig (and you can do that too with the theme_tweaker) if I add at the end of it

{{ this.order.profile.membership.name }}

it will show the membership name. Keep in mind this is the membership at the time of order. Changing the membership of the customer later will not affect this order's profile.
If you want to grab the current profile membership regardless of order use

{{ this.order.origProfile.membership.name }}

Also neither one will show name if there is no membership assigned to the profile in question.