I have not personally had to make this change, so I'm not speaking from experience, but looking at the documentation I would try the following: (you could cut corners and avoid some work by modifying the existing modules, but this is the cleanest way I can think of)
- create a module that stores your field and the the order number.
- using from_alter modify the invoice editing screen to give your users a way to edit the field.
- use the invoice template to pull in your field, based on the order number. invoice "system" documentation.
It's not a simple change, but using this pattern you'll get the modification you are looking for without having to worry about updates not working.