It turns out you can trigger a 2FA verification code by going to:
iOS10: Settings > iCloud > Password and Security > Get Verification Code
iOS9: Settings > iCloud > [Your name at the top] > Password and Security > Get Verification Code
Now the challenge here is that the original iPad Retina was limited to iOS9.
The other challenge is that the iPad retina running iOS 9 appears to have a 4-character 2FA - whereas my other devices are generating 6-character verification codes. I'll try and solve that somewhere else.
There appears to be a trick to get the verification code sent by SMS.
Edit - it turns out it was asking me for the login passcode, not the iCloud verification passcode. Going to give the tick to the other answer to thank them for helping out.