Can I use a CFL or LED bulb with a higher lumen output as long as I stay under the maximum wattage listed on the fixture?
Yes. The more efficient bulb will also produce less heat, so it should be fine. The only concern is that if there is a dimmer you need a dimmable bulb, and if there is a touch switch or other circuit that was powering itself via a trickle current thru the bulb that may not work -- but neither it nor the bulb will be harmed by the experiment, so go for it.