Since there is no Carbink friend safari, how do I actually get a Carbink with its hidden ability, sturdy?
You can get a Carbink with it's Hidden Ability Sturdy in Reflection Cave, in a Horde Encounter.
Pokémon found in Horde Encounters have a small chance of having their Hidden Ability.
Hope I helped :)
You can get it by encountering them in Horde Battles. Carbink Horde battles can be found in Reflection Cave. Some will have a small chance that some Carbinks have Sturdy.