its not important to have chinese name on your business card. but i would recommend u to include both ur Chinese name and your name

usually my business card has the chinese version on the back... it will help in the communication and generates some funny comments.
Also, if Chinese use foreign names, why shouldn´t we use the Chinese version for ours, if available (mine is 大卫)? it will make it easier for them, and avoid being called the "wrong thing".

Remember where you are. It will add to client base not take away from it

Yes I think you should have a Chinese name on your business cards, my name on my business card is "鬼魂“