I made mine about 4” high and used metal window screen for the bottom. I lifted a outer cover on a really cold day last winter and the shavings were wet on top but just below the surface they were completely dry..
Now to figure out how to implement a boardman feeder with it.
Mine are also 4" high. I have a screened 1-1/2" by 10" vent slot on each end 1" down from the top edge. The telescoping cover hangs down to cover part of it, I center the telescoping cover. I used to use them with wood shavings and my experience was the same. If you put your hand down into the shavings the shavings on the bottom were warm.
Last winter I converted three of them to Vivaldi boards. I cut 1/2" plywood to the inside dimensions and used 3/8" shims to position it and then secured it in place with pocket screws. I was so pleased with how well they worked and how much easier they were plus the ability to add sugar without opening the hive that I converted my other quilt boxes to Vivaldi boards over the summer.
If you don't have the tools or skills to build a 4" height box then you can use a medium or shallow super. The medium is 6-5/8 high so its not a lot larger. The shallow super is 5-11/16" high.
Shallow super height is good for shavings quilt box and also will serve to house the Mann Lake hive top feeder inserts or if you wish do shallow cut comb. Make that piece of equipment serve multiple uses.