Clear silicone? Kinda lossy gasket.

Ju are you not going to paint the inside? this may help prevent the MDf from absorbing moisture from the air.

PS I know three spikes looks better on paper and is easier to level, with young children around, I always go for 4, 3 way are prety unstable if you push from one of the apexes. Maybe silicone would be a good idea as it would lower the centre of gravity, as your box looks top heavy.