Well this comes as much as a surprise as it doesn't, Bozai has specialized dialogue for when he can't give you snow boots due to full armor inventory : r/Breath_of_the_Wild
Bozai - Zelda Dungeon Wiki, a The Legend of Zelda wiki
How To Get the Snow Boots in Zelda Tears of the Kingdom
Better Sand & Snow Based Armor [The Legend of Zelda: Tears of the Kingdom] [Mods]