Very important that any temperature change for bulbs (or any plant that should be stored over winter) goes slowly. This is the way we store frozen plants over winter at the tree nursery that I work. The earlier the plants are lifted, the longer the temperature change needs to take.
Some insulation over that box seems very wise to me if there is risk that the bulbs will freeze too quickly. Eventually I imagine you will need to add a lot of insulation.
Otherwise a nice place to overwinter somewhat tender plant material at home can be to use an open leaf compost. Easy to dig in even if the ground has been frozen for some time. It's moist, but there is always plenty of air and as it's above ground there can never be standing water. I often dig pots directly into the compost for overwintering if I run out of room in the greenhouse.
Obviously any bulbs outdoors in such a protected environment needs to be protected from rodents and the colder the climate, the larger the composts needs to be. Or one could dig under the compost and store the bulbs there, but then they may need some protection from too much moisture, depending on local conditions. On the other hand a compost with too much greenery added could actually be too warm. Still for anyone with a leaf compost it can be well worth trying. A leaf compost plus a thick layer of snow would provide some serious insulation!
This method is quite similar to where you dig a hole in a well drained spot and wraps vegetables and potatoes in an insulating layer of straw or similar, covering it up with a layer of soil, adding more insulation as it gets colder, but it's much easier to implement. Not as fancy as an earth cellar and not as easy to access during the winter, but this isn't a problem for bulbs.
