A different shoe (or sock combination) on each foot is EXACTLY how I do it. This is the equivalent of a scientist using a control group in an experiment. Yes, you need to switch your "control group" around occasionally; the experimental foot becomes the control foot and vice versa. I would not recommend one pair on the way in (when your feet are fresh, say you're mostly climbing) and the other on the way out (when tired, trail going down). Too many variables interfering. You need to switch back and forth several times during your trip to average across variables (such as tiredness, slope, etc). So yes, you'll have to take both pairs with you and carry the unused ones. A minor sacrifice on the altar of science.
I've never had this method throw my pace off or give me troubles. Just the opposite, it helps me prevent troubles. You can really hone in on relatively minor differences this way. When you find an improved set of sock/shoe, that becomes your new baseline, and your new control for the next experiment.
Who cares what it looks like. Science isn't fashionable.