Sounds like this might be your area of expertise. For the rest of us, take a shoebox. How much ballpark extra weight we talkin’ to have a livable planet? (Maybe the mushrooms would be ~2x as heavy as standard shoeboxes for example, to meet existing spec.)

Or how about for the glasses box they show on the site in OP, or a plastic sleeve like Americans sell Oreo cookies in. Anybody have any guesses?

I've done some experiments with mycelium as a construction material, but I'm hardly an expert. Mycelium weighs anywhere between 50 and 950kg/m3. Usually you won't have mycelium as thin as cardboard, because you want use mycelium as a 3d buffer, replacing styrofoam. EPS (styrofoam) has densities of 15-30kg/m3. So while it's more sustainable it's also heavier.

Hopefully not used as packaging for Oreos, because unless the fungus has been highly adapted to the substrate, the mycelium will try to grow into the food. Oyster mycelium won't be toxic, but I don't want my Oreos to taste like mushrooms.

Heavy means more fuel to ship it. Maybe still a net-win, I don't know.