Here is the full list of all 4 Hard West 2 Playtest achievements.
Finish the Hard West II beta.
Finish "Fooling The Hangman" in no more than 4 turns with all objectives completed.
Finish "Daredevils" in no more than 8 turns with all objectives completed.
Finish "Blasting Out" in no more than 8 turns with all objectives completed.