I've done everything but I can't find Louis Letrush at either the inn or near the stables. Can I go ahead and keep Frost anyway?

Louis Letrush is not supposed to be neither at the inn nor at the stables at the end of the quest. He is supposed to be at another location, in the wilderness of the Rift not far from Lost Prospect Mine. You should try to search for him there, or simply turn on the quest marker.