Since you mentioned speed is more a priority then cost here is what I would try.
First, I think sanding with 400 grit is probably causing more work then necessary. There is no doubt that you can level paint quite quickly using such an aggressive grit, but it often requires a lot of follow up sanding with subsequently finer and finer grits to remove the damage.
When leveling orange peel (texture) from the surface the backing plate/sanding block is just as important as the grit. I would start with something like Meguiar's Unigrit 1000 grit sandpaper wrapped around the Meguiar's Sanding Block. The sanding block is fairly firm and will do a good job in helping to level the paint quickly. The 1000 grit paper should have zero problems leveling fresh paint unless it is extermely hard.
Note: With the sanding block and 1000 grit I would sand in one direction, maybe front to back-back to front to keep all of your sanding marks uniform.
Next I would sand with 2000 grit Meguiar's Unigrit at a right angle to your original 1000 grit pattern. This is called cross hatching and it makes it very easy to tell when you have removed all of the 1000 grit marks.
For example: Lets say you sand with 1000 grit left to East to West-West to East. All of your sanding marks will run that direction (East and West). When you sand with the 2000 grit run it North and South. As you look at your finish all sanding marks should now run North and South, if you see any that still run East and West you know you need to sand that area a little more.
After you finish with 2000 grit you
could probably go straight to polishing or you could sand with a subsequent final step of 3000 grit.
-With 3000 grit you could hand sand using Meguiar's Unigrit or machine sand with Meguiar's 3000 grit finishing disks. I prefer to use a machine to DA sand for the final step as I feel I get a more uniform microscopic texture in the paint.
What kind of machine are you using to polish?
Meguiar's 1000 grit unigrit
Meguiar's 2000 grit unigrit
Meguiar's 3000 grit unigrit 
