Based on that same concept, do you have any sheet metal that you could form into a cone? Or, you could try cutting a hole into a piece of wood, setting it down like a table, putting the sphere in that, and then drilling. With these ideas, the force from you pushing down would be met with the force of the table/metal cone/egg carton, which would put the pressure away from the walls that you're creating.
It would also be better if you could line the hole/point of contact with something soft like weatherstripping or foam to get as tight of a fit as possible, ensuring that the sphere wouldn't move(with the egg carton, because it's so flimsy, you wouldn't need this).