"Hey, that's why they made PhotoShop!"
No, that's why they make light meters and fill lights!

"The grey card is to determine white balance."
Not true! The grey card is calibrated for only one thing - 18% grey. A grey card IS NOT color neutral! A light meter can only determine one thing - 18% grey. If you take your meter reading off of the grey card, for that given light, aperture setting, shutter speed setting and ASA (oops, I mean ISO), you will render that 18% grey tone correctly (This is also known as Zone 5 in the Zone system). If you want that tone rendered lighter or darker - that's when you change your camera settings.
For accurate White Balance - you need to use a specially made white balance target - such as those made by X-Rite. This target will be color neutral. A white sheet of writing paper is designed to look white - but in reality is made up of different components, such as clay and titanium brighteners - that usually have a slight color bias - purposely designed to be imperceptible to the eye, although they are not imperceptible to the camera! If you set your white balance using a color biased target - you will get color biased images! From the X-Rite website description of their "Colorchecker White Balance Target : "the ColorChecker Custom White Balance target is a scientifically engineered, absolute neutral white reference that prevents color shifts and provides precise, uniform surface that is spectrally neutral – in any lighting condition."
X-rite sells a small "ColorChecker Passport" that contains white balance, grey targets and color targets - all in a pocket size package for $99!