That's what I'd do too, except I would also change out of "biking clothes" to office clothes because of the sweat on the biking clothes. For me, "biking clothes" would just be a moisture wicking shirt (doesn't have to be a "cycling jersey", although I just found out about cycling jerseys that don't looks so obviously sporty) and shorts.

8 miles cycling in San Francisco would definitely cause me to sweat buckets, even if its under 50 degrees. Climbing really makes me sweat.

I would guess that the best way to improve your commute time is to improve your climbing speed. I don't think wind resistance is much of a factor in climbing, and reducing wind resistance (on flat ground) is the reason people put on sporty cycling clothes. You'll of course go fast on descents (unless you're riding a fixie - no experience on those myself, just heard you can't coast on them) but you might be restricted by traffic stops and the like.