The cheapest way to get your bike lighter is to ride it a lot and get in good biking shape. 3 miles at a time isn't much at all once your body gets used to it. It isn't even about leg strength as much as it is about your body learning to do something, adjusting to it, and becoming more efficient. My first bike was a steel frame full suspension $69 jobber that easily weighed 40 lbs. The first rides the wife and I did were around 5 miles. After a month we were doing 30-40 miles at a time and 5 miles was nothing.
A dense downtown with lights every block will kill any time saving efforts. So shaving 10 minutes on a 3 mile ride is highly unlikely unless you are going brutally slow right now and can make up some ground with riding a little faster. If you get to a point where you can time the lights, that can help. There's a section on my commute where I'm sprinting to hit a series of green lights. There's another section where I ride slower to hit a couple.