Hi Robert,
You would see an "AdWords API Center" tab only in a production manager account and not in test manager account. Test accounts (both client and manager) and production accounts work in separate domains and cannot be linked. When you log in to a test account, you would see a
red banner on the top of the page indicating that your account is a test account. Could you please check if your account is a test account?
If that doesn't help, could you please provide me with your 10 digit account ID so that I can look into your case further? Please click on Reply privately to author when responding.