Maybe this belongs on the Indian call center thread. It seems silly that they couldn't do this for you.

Some things you can try:
1. Try doing it yourself on-line on the Amex website. It might work without human intervention.
2. See if you can use a middle initial on one card and your name without the initial on the other.
3. If you need to go through a phone rep, don't specifically mention the new card is for the same person. Other than the social security number, which they might not check, there's no way for them to know.