i think you should build a box that fits the required dimensions and go down to the hardware store and buy a can of "great stuff"

 

its a foam that sprays out of an aerosol can. when it sets it expands and then hardens. fill the box most of the way, add the egg, cap off the top and let the foam set for an hour.

you're right in saying it's the mass that effects the inertia, but the weight is the factor here. to decelerate a 1kg mass moving at 1m/s you apply a force.

if the object already has bouyancy acting on it, say, 900g upwards, you only need to apply a tenth of the force to the object itself because the rest is due to bouyancy.

if your egg is neutrally bouyant, you could describe the egg as though it were just another bunch of the fluid. the tub decelerates suddenly applying the full stopping force to the egg via bouyancy.

so the egg is suddenly in an environment where it's x distance underwater and the gravity is equal to G + deceleration rate. if the egg has positive bouyancy, it'll actually shoot out the top of the tub as it's displaced by the water.

the egg, under that apparrant gravity might not survive the pressure. which is the only problem. if you can drill into the pocket of gas and fill it with something incompressible before you toss it around, it will then survive anything the tub can handle.