You can die cut any quantity of cards BUT the setup costs make it uneconomical on less than a few thousand. I used to work for a foil printer who did custom die cutting, while I was at college (early 2000s) and even then you were looking at in excess of £50 for a simple die. Plus around an hour setting up the machine (we used a converted Heidelberg Windmill) then the length of time actually stood operating the press as it cut (often as little as 15 mins for a few thousand impressions).
A die would normally cost about £50-60 and a print finisher would have a minimum charge of about the same....therefore you are looking at adding £100 + to your print bill. The cards would probably have to be litho printed for the registration. If you are really wanting a price I would suggest going to a local litho-printer and asking.
Wouldn't it be better just to have a normal business card with the graphic detail on the card. Sometimes its better to conform to convention but have your innovation on the standard size and shaped card. Look at the practicalities.
Not trying to be a 'party pooper' just trying to help.