I spent a month of weekends and some weeknights wet sanding the car by hand, with 1500 grit paper. When you get to this delicate a grade of paper, the only thing you need is elbow grease and a lot of patience. I spent around 6 hours on each panel.
This is what the surface of the paint looked like after multiple coats of paint and clear.
And this is what it looks like after wet sanding. Dull, right? So the next step is polishing!