In last night's search I did find business cards with rounded corners. Can't remember if they were for laser/inkjet or a printer.
I do remember one time emailing a printer and asking if they could send me a sample of a business card that would be like a playing card, and what I got did not feel and handle like a playing card.
Very brief search of USPC website did not turn up any custom business cards.
It does look like the good folks at Card-Shark have all sorts of custom playing cards with a good-handling finish, and can do custom jobs: