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.