So if I'm going to be carrying this amount of stuff either way, will a rack weigh as much as the entire rest of my bike? Or is this just Something You Don't Do?

Unless your bike is *really* light, it doesn't seem like the rack weight should matter that much. (I have no actual data, but as reference to what I'm trying to say, I can't tell if this is a serious question.) I'd go with the rack, if your bike has the eyelets or whatever for it.