Maybe there's something special about your job that would prevent this, but if it's possible to bring them in the day before, have you considered just leaving a change of clothes at work permanently?

Shoes - can never be washed really (with a few exceptions for especially dirty jobs).
Pants - usually only wash them once a week.
Shirt - I don't wash my shirt every time I wear it. I just find that it doesn't matter - what matters for smell is always showering. Of course I could just be lucky in the body odor department. But even then, you could leave multiple shirts at work - or *just* carry a shirt on your bike.

As for what to carry them in - just put them in a trunk bag that goes on your rack. Here's one -
http://www.amazon.com/MTX-Trunk-rigi...ords=trunk+bag

Here's a cheaper one -
http://banjobrothers.com/products/cu.../rack-top-bag/

Using bungee cords is just a disaster waiting to happen - I tried that once, my jacket sleet worked itself loose and caught itself in the rear tire. Fortunately it didn't bring the bike to a halt and crash, but it did rip a big hole in the jacket sleeve. You said it would be in a bag, but the bag can come off...anyways, point being, they make bags that are specifically designed to attach to a rack, and they're more reliable than using a bungee cord.

But mostly, unless you're actually commuting every single day of the week, the best solution is to leave an extra set of clothes at work so you don't have to carry them back and forth.