I've had plenty of carbon and aluminum bars. Whether it is worth it to spend extra on carbon really depends on the bar. There are good and bad carbon and aluminum bars.
What is more important is that the drop, reach and bar shape fits your needs. Also, note that a lot of aerobars flare at the drops. So a bar that is 42cm wide at the hoods can be 44cm wide at the drops.
And if aero is your goal:
- Narrower bars will likely have a greater impact than aero shaped tops.
- Carbon bars likely have better shaping than aluminum.
Lastly, there is nothing wrong with using an aluminum stem on carbon bars. Just make sure you use the correct torque specs on the bar.