But you don't need to 'dissipate' the energy of the egg. All you need to do is to limit its acceleration so that the forces aren't too high. There is an essential difference between the two ideas. It really wouldn't matter what actual speed the egg reaches during a bounce as long as it is not subjected to an accelerating force that could break it.
In your diagram you show a hollow shell that is a sphere when not under stress? But you now imply that the ball will be almost full of foam. I was assuming a rigid sphere (stronger than your pictured bungee design) which would have spokes going from the centre to the edges. The spokes (radials) would be long coil springs, supporting the egg carrier at the centre. These springs would allow the egg to move freely under moderate acceleration whilst the sphere is doing its violent bouncing.
Your main trouble will be in making a bouncy ball that is actually bouncy enough. In order to be bouncy it must not flex very much because flexing will involve energy loss due to hysteresis.
Have you seen the sort of sphere used in 'Zorbing'? That was the sort of structure I am envisaging - with the egg supported like the human is, only with springs instead of the straps.