Travis is certainly disgusted with Eckels because of his cowardly reaction when he first sees the Tyrannosaurus Rex and he and Lesperance are, therefore, obligated to shoot the monster. But, Travis threatens to leave Eckels unless he removes the bullets from the dinosaur for two reasons: They cannot be left behind as they are anachronistic, and Travis is disgusted that Eckels has stepped off the gravity path.
Eckels complies because he has violated his contract with Time Safari, Inc. and, also, because he does not want to be left behind, and he may be hoping that the guides will not report his violations. After all, Eckels has been told,
If you disobey instructions, there's a stiff penalty of another ten thousand dollars, plus possible government action, on your return.
Also, since Travis and Lesperance have truly saved Eckels's life because he is the one designated to have taken the shot, his failure has allowed the monster to advance upon them; so, there is no excuse that Eckels can make for not at least retrieving the bullets from the dead dinosaur. He owes his life to the guides.