Chapter 3 of AFM doesn't mention anything about prop lever setting during engine restarting for both windmilling as well as stationary propeller
If your engine fails in a DA-40, you've lost oil pressure, so there likely isn't enough to push the blade to the coarse position.
A DA-40 doesn't have an accumulator like a Seminole, for example. So the blade position is already a fine pitch before cranking the engine during start, and the oil pressure allows the blade to be pushed to the coarse position, not the fine position, which is also the reverse of my exemplary Seminole, which needs the oil pressure to maintain a fine pitch. The Seminole's shutdown-default is course for the same reason the DA-40's is fine: a lack of oil pressure.
The likely reason the emergency procedures don't specify, is the system isn't designed to allow you to feather or even coarse the propeller without the oil pressure generated by the engine, not that a DA-40 can feather at all, and you wouldn't have much oil pressure if the engine failed.
At best, the windmilling propeller would turn the oil pump, which would provide some oil pressure, which would in turn coarsen the propeller to some degree, but as soon as the propeller RPM drops further due to the resistance generated by the coarser pitch, so would oil pressure with it, and that would lead to the propeller going toward a more fine pitch again.
To maximize the chance of getting a good restart, allow the propeller to windmill by placing the propeller lever at the fine pitch, high RPM, setting.
Here is a diagram of the system hosted online. You can see that added oil pressure to the piston causes a more course position of the propeller.