It’s possible to get fairly accurate tangential curves to circles using some 45° rotated squares as guides, i.e. squares that are the same size as the radius of each circle, rotated around the circles’ centres using the Rotate tool, and then drawing a path with the Pen tool, making sure that the bézier handles of the curve snap to the corners of the squares.
Then you can copy and paste the path you made, and reflect it 180° and move it so that the ends snap to the centre of the circles
Then once you have your shapes, you could use the Shape Builder tool to make the pieces.
And finally you can use the Direct Selection tool to select all the anchors of one side, and freely move them.
Or even scale them
Here’s a zoomed in shot showing the accuracy of the tangent, in outline view.