I am shooting in natural light, with my subject facing a full floor-to-ceiling south facing window. You would think my meter would allow ISO between 200-400 in bright light, but I was lucky to get it to 640, with f/3.2 and 1/60 (manual mode). I've read other places that people say put your ISO down to 100 for indoor shots for greatest clarity (especially sharp eyes) — but don't know how I can with subjects potentially moving. Is there a setting in my camera (Canon 5D Mark III) I am missing, or something that might be making my meter need more light?
It would take a rather brightly lit room to get your ISO down that low. I've got a low hanging, 5 light fixture in a small white room and I just metered f/3.2, 1/60 and ISO 1250. So, bright sun is definitely going to help, but ISO 200 or 100 inside, at f/3.2, without flash is an impossible dream.
You either need a faster lens (like a f/1.4) - but that is going to require some razor thin depth of field probably by setting it really low. Or you need a flash. If you haven't used a 'real' flash before - they have a bad rep till you get to know how to use them either bounce-style or off camera.
If you really want to shoot that low ISO inside, get a flash. Its well worth the investment.
But frankly, a Canon Mark III shooting at ISO 640 is completely fine. The noise really should be fine for a Full Frame camera at 640. Of the things to worry about, I wouldn't cry too much about it.
Because the scene is darker than you assumed it would be. Indirect sunlight is not as bright as direct sunlight. It is usually nowhere near as bright. Much, if not all, of the light that is shining through the window has likely already been reflected by something else and unlike when you are outdoors, you're not getting any reflected light from directions other than the direction the window faces. Think of the window like an aperture. It is restricting the light entering the room to one direction. While it is allowing light from the south at specific angles to enter the room it is also not allowing light from above, below, east, north, or west to enter the room.
It is not the meter that isn't allowing you to shoot at the ISO, aperture, and shutter speed you would prefer, it is the amount of light that exists in the conditions under which you are shooting. The meter is only telling you how much light is present in the scene.