The only reason I could see to put a cover over the tray is to keep light away from the roots. If your root area is properly ventilated and properly light sealed this is a very good route to go. As for the drip system idea, You could implement this without needing a lid but you do want the roots to be out of the light. You need to rig up watering lines (found at local hydro shop or hardware store in the sprinkler section) with drip nozzles or halos at each plant stem. You will use quite a bit less water since you will have one main reservoir that the water is coming from and going back into.

You might need to rig up a special line that goes from the bottom of the ebb and flow tray to your reservoir. This is where the gray area is for me since gravity plays a factor as with other physics and properties of water.

At the very least your going to need two pumps and tubing.