French refrains from killing the moose, but he kills Travis.... why?

French is devastated beyond reasoning because of RiRi's death. He momentarily refrains from shooting Travis, but when he thinks about the fact that RiRi is lost to them, and that it is Travis who handed Wab over to a gang, in addition to the fact that Travis will killed likely to keep turning over Native people to Recruiters..... he shoots him.