This is actually something which the Shape Builder Tool makes very easy and you can use the circle configuration you already have....
Without the need to draw any more paths.
- Simply select the circles and grab the Shape Builder Tool
- Click and drag on the areas you wish to combine.
- To remove an area hold the Option/Alt key and click-drag.
This makes it a matter of seconds to get from the circle configuration to the desired end shape.
Note: This assumes the original circle configuration has the circles actually touching one another. If there's a slight gap between circles it may result in uneven curves in the final shape.