i would use an entrance reducer.....screened bottom boards are not going to keep mice from entering the bottom deep/medium of a hive through the 'front door'.....
entrance reducer for 'winter chills'
i do place entrance reducers on, the wide side, 3 inch width. i do it as soon as i take honey off and just keep it that way through fall and winter months.
mice can chew through wood, or squeeze through the small opening (industrious little critters that drive me crazy). as far as winter chills?
less cold air whipping through the entrance of the bottom board is probably a good thing.