Hi XeXE,
However I want to change a phone number in there, but I can't figure out how to modify the message template that the system is sending out in emails to change that phone number.
I'm a little confused about your request:
Do you want to change the value of the phone number, or do you want to customize the phone field on the document?
(1)If you need new value of phone number, you can modify the phone value in the form directly(add the phone field in the form).

(2)Customize phone field in the document:
The template shown in the dialog is from word template:

If you want to show new layout for the template showing, you can create and design new word template as your needs:
Use Word templates to create standardized documents - Power Platform | Microsoft Learn
Could you provide screenshots about it? which helps me to better understand your needs.