Make a room within a room. Don't let it touch too much of the structure of the house suspend it from the ceiling if poss (on elastic)
line it with pyramidal sponge cones a la an anechoic chamber.
It is very cloying inside such chambers so keep singing.
In truth the chamber need not be that much bigger than your head but I doubt you could stand it that small.
make it out of cardboard for lightness. It need not cover you body as long as the floor fits you torso approx
If there was any way to amplify your voice and listen on headphones you would get the aural effect if not the right oral effect.
tell us how it goes (quietly please **BG**)