Really interesting idea and I can see many game mechanics that benefit. What do you think is the minimum thickness using any technology available?
If you are willing to spend even more, you could bundle the card circuitry with the silicon embedded in the e-ink display already, and there are FPC e-ink displays. So it'd be about half the thickness of a credit card and flexible!
(at that point, we'd need to change the plinth interface, probably go for NFC like people are mentioning in these comments)