How do I find an unidentified expansion pack's product code?

Does the game still show up on your Origin client in myGames ?

You don't really need the code again as the game should be on your account, ready to be downloaded again

If not, try the following things (test if it's there after every step)

• Log out and back in and make sure you use your email address and not your username to login.
• Restart Origin
• Reinstall Origin
• Restart your router
• Contact a game advisor (see below)

Game advisors can be found through live chat or call-back

1. https://help.ea.com/en/contact-us/ 
2. Click on the orange NEXT button
3. Select a platform
4. Click on the orange NEXT button
5. Select a topic and describe your problem in a few words
6. Click on the orange NEXT button
7. Click Chat Now to chat with a Game Advisor or leave your phone number for a call-back
   

In Details here: How to contact a Game Advisor using the new EA Help page