Sorry this happened to you. I used to work at a large bank and unfortunately we had multiple cases of fraud daily.
Regarding NOT putting your CC info on your KTTW card and using your credit card directly. I think I would feel safer putting my CC info on my KTTW card vs using my actual CC each time for the following: Your actual CC leaves your hand one time during the trip and that is to hand it to the CM upon check-in. Sure the card number could get stolen as it sounds like this could be the case with the OP. But, each time after that, the CMs handling your transaction don't have access to the number. I would think (and could be wrong so someone correct me if I am) that with your KTTW, that since it's all computerized, the CMs would have no access to your actual CC number, expiration date, and security code on the back. But, each time a CC is handed over to a CM such as a waiter, they can write down all the necessary information to make a fraudulent card as they now can clearly see the card number, expiration date and the security code. Every time I hand my card to a waiter, regardless of where I am, I fear of that happening. But, my only other option would be to carry large sums of cash around which I refuse to do. As I use to tell customers at the bank, if you loose your wallet full of cash, you can kiss that money goodbye. If you loose a wallet with card(s), sure it may be a pain but with the unauthorized charge protection that credit cards offer, you will get your money back.