Fitzgerald sure had some chances to kill Bridger, but why should he?
Once he got Bridger to buy his story there was just no need to get rid of him. You have to remember that they still had quite a journey to go to reach the camp, with the Ree still on their trail. Bridger was of much bigger help as an active part of his company than just undertaking this journey alone would have been. Especially once he got Bridger to go with their story on the one hand by convincing him that there was no way to save Glass and on the other hand by guilt-tripping him into not revealing the whole truth either, there just was no need to kill him.
And add to this that a Bridger (who the commander knew was an honest guy who cared for Glass) who confirms Fitzgerald's story gives much more credibility to it than if Fitzgerald (who the command knew was a rascal who didn't give a damn about Glass) had just returned alone claiming that everybody else died an honorable death.
In addition to that, while Fitzgerald killed Hawk that wasn't entirely planned but more of an accident in the heat of the argument (and even then he was just some "filthy redskin" for him anyway). He did want to kill Glass, but in his eyes Glass was pretty much dead already anyway and just a corpse they had to guard for nothing. As much of a bad guy Fitzgerald was, he wasn't just going around stabbing the people of his track from behind, especially when they're still of use to him.