Is there a good way to make a composting watering tank that won't clog 1/16th inch holes?
What you're looking for is called compost tea. The compost is placed in a barrel of water and aerated, then the solids are filtered out. The resulting water can be used in an agricultural sprayer to fertilize fields. Harvard uses and promotes it extensively if you're looking for more information. 

At home sized scales you can use aquarium pumps and aerators, along with pantyhose filters.