Physics behavior - Apply force towards position. The position would be the end of the bungee cord/attachment/origin point. Use distance from object to origin point, the farther the distance the greater the force applied, to create the bungee effect.
As for the bat, you can check collision or overlap between the bat and bungee object, then Apply Impulse Towards Angle. Impulse is how hard you want the hit to be, and angle you can use the angle between the bat and the bungee object.