Although one may initially assume that when Doug goes back in time he is still operating in the same timeline he left, this is not the case. There are actually 4 individual timelines in the story, only in the last of which Doug succeeds in preventing the bombing.
In Timeline 1 Carrol uses his original car to blow up the ferry. Claire is not harmed. Doug sends the note back in time, creating Timeline 2.
In Timeline 2 Doug's partner intercepts Carrol, who kills him and ruins his car. Thus Carrol kidnaps Claire, cuts off her fingers and burns her alive at the warehouse, using her car to blow up the ferry. Claire's body is not wearing a red dress and there is no message for Doug on the fridge. Doug then goes back in time creating Timeline 3.
In Timeline 3 Future!Doug steals the ambulance, crashes into the warehouse and rescues Claire. They go to Claire's apartment, she changes into the red dress and he leaves the fridge message. However he then makes the fatal mistake of leaving Claire at the apartment—Carrol, knowing where she lives, shows up, burns her alive and cuts off her fingers. Future!Doug is unable to stop the ferry explosion and dies in it. The events of the movie then begin here.
In Timeline 4 Future!Doug does everything he did last time except this time he takes Claire with him to the ferry. They are successful in preventing the bomb from going off on the ferry but Future!Doug is killed. Claire then encounters the Doug from this timeline.
At the end of the movie, Larry's still dead, but nobody in the world knows it, because the only people who knew about his death are either dead or in another timeline. We also know that he won't be missed for some time, thanks to his conveniently timed vacation. This may lead some people in that timeline to eventually wonder "whatever happened to that guy?"
But the bomber would be identified relatively quickly, thanks to Claire, and his compound searched. This would lead authorities to find Larry's remains. They'd also find Larry's DNA inside the Blazer and the slugs from his service weapon. The only mystery would be how he ended up dead.