So my experience is a bit different than the OP but might still be relevant. I have had the Amazon Visa Card for awhile now. I was automatically upgraded to the benefits of the Prime Visa Card when it was announced in January (been a Prime member for years). I called Chase shortly after the announcement of the new benefits and the metal card to see if they could send me the new metal design. They told me that I had to wait and all existing card members would recieve the new design later this year automatically, but that my current card would get the new 5% back benefit (along with the others perks, such as no FTF).
I waited two months before sending a Secure Message to Chase via my Chase.com account (this was about 2 weeks ago). I simply asked if it was possible to get the new metal card because I am a Prime member and I knew other people that had already recieved their metal cards. Chase responded within an hour telling me they were rushing the card out to me. I was surprised to see the card in my mailbox the next day, delivered FedEx Overnight Shipping.
Very impressed with Chase there. However, due to my vastly two different experiences between my first call and my secured message leads me to believe that YMMV. If one CSR tells you no, just try again, and hopefully you eventually get an awesome CSR like I did who will overnight you the card. 
My benefits were upgraded on Amazon right away but the new card was never mailed. I finally just requested a replacement card and the new metal card came.
