Hi,
This is already the case. The selected choice should appear in the invoice, order details, etc.
If you don't see it, it's likely because you have custom CSS code which hides it. You probably added your CSS to hide it on the product page but your CSS is not narrow enough to target only the product page and thus it affects other pages where you want it displayed.
Now, that's just a theory, because you didn't provide the URL of a product page, so I can't confirm the situation on the website. But based on what you said and what I know, it's the most probable.