Extra text can be included on the shipping label using package options on the Customer Options page.
First create a package option template with the text that you want to appear on each package for that customer:
Note that FedEx and UPS use different fields. Other carriers may use other combinations of these 4 fields.
See this article
for standard placeholders that can be used for the text, such as invoice numbers, etc.
Then create a customer option to select the package option template:
This text will then be automatically included in any new shipments: