I'd think of these as 'platinum' badges and can be obtained after gold badges.
-A badge that can only be earned as a living character
-A badge that can only be earned as an undead character
-A badge that can only be earned as a dual-type character
-A badge that can only be earned with a dual character
-A badge that can only be earned as a shiny character
-A badge that can only be earned as a Fakemon
-A badge that can only be earned as a mutant character
-A badge that can only be earned as the base stage of evolution
-A badge that can only be earned at the second stage of evolution
-A badge that can only be earned at the third stage of evolution
-A badge that can only be earned as a Kanto-native Pokemon
-A badge that can only be earned as a Johto-native Pokemon
-A badge that can only be earned as a Hoenn-native Pokemon
-A badge that can only be earned as a Sinnoh-native Pokemon
-A badge that can only be earned as a(n) Unova-native Pokemon
-A badge that can only be earned by a character brought back from the dead (with the revival ticket)
These are all just suggestions, you don't have to use any of them, but they go with the theme of 'RP as this to get a specific badge'.