I read where you got some sprinklers. Why not make an Aeroponics system then?

As for your question you can use a diverter to feed the drippers and recalculate the water at the same time. It's the movement of water that I feel is important. Think in nature it safe to drink from moving water but when the water is standing in a pool bacteria starts to grow and you don't drink the water. Same idea with your plants.

I do like using submergible pumps. Yes it does raise the temperature of the water a little but it's a fair trade off to me. In most fish stores or pond stores you can find diverters that allow for multi water out put from a single input. Make one to feed the drippers/sprinklers and one to act like a jet under the water. Just when you start an Aeroponics system make sure you place the roots at the bottom of the net pot and try to get some of the roots to start hanging out the bottom before you add the grow rocks.