If you watch a lot of heist movies, you may recognize Bigwig's type: the big, strong guy (er, rabbit), the one whose job is to fight. He's the enforcer, the hired muscle, the goon. Yeah, except, in a lot of those heist movies, that guy is working for the enemy. And that's what's so interesting about Bigwig and why we're giving him second billing in our Characters section. Sure, Hazel is awesome pretty much from the beginning of the story (good for him). But Bigwig starts out almost as a potential bad guy and becomes awesome by overcoming his flaw.
Bigwig's flaw is captured in his name: a "big wig" is an important person. (And yes, the phrase "bigwig" comes from the old style of wearing big wigs if you were important (source.) Since he thinks he's an important person, at the beginning of the story, Bigwig isn't always easy to get along with.
We see this issue brought up several times at the beginning of the book: for example, Bigwig wants to leave Sandleford Warren because the Chief Rabbit was mean to him. Bigwig even knows that other rabbits would act differently: "I dare say a good many rabbits would have kept quiet and thought about keeping on the right side of the Chief, but I'm afraid I'm not much good at that" (3.12). Now if you put it that way, it might sound positive: Bigwig isn't going to suck up to the Chief Rabbit like a lot of flunkies might. But looked at from a different angle, it means that Bigwig isn't very good at getting along with others even when that's the smart thing to do.
That's a quality that Hazel and the narrator come right out and tell us about. Hazel thinks that Bigwig might be useful, but "he would also be a difficult one to get on with. He certainly would not want to do what he was told—or even asked—by an outskirter" (4.20). This isn't the kind of dude that you want on your military expedition/heist because he doesn't play well with others. He is, in short, not a team player. Hazel only manages to deal with Bigwig by being very cunning (see "Themes: Cunning and Cleverness").
Bigwig's "me-first" attitude is especially apparent at the beginning when it's a question of protecting the smaller, weaker members of Hazel's group. When Hazel's group is between a river and a dog, and the little rabbits Fiver and Pipkin can't swim, how does Bigwig react? Bigwig simply says, "Those who can swim, swim. The others will have to stay here and hope for the best" (8.29). In other words, Bigwig thinks everyone should take care of themselves. If Fiver and Pipkin can't swim, well, tough patootie.
And that's how Bigwig feels… that is, until he gets caught in a snare in Cowslip's warren and everyone pitches in to help save him (18). This is a painful lesson for him to learn (mainly because he almost chokes to death), but everyone pitches in and helps: Bigwig knows a little bit about snares; the big rabbits do some serious digging; and the little rabbits bite through the peg where only they can fit. In fact, Bigwig wouldn't be so close to death if he only listened to Fiver in the first place. So if we were looking for a moment in the book where Bigwig learns his lesson (that little rabbits are important, too), this would probably be it.
In fact, just to make sure that we understand this lesson, the narrator makes a big deal of it in the next chapter:
Since leaving the warren of the snares they had become warier, shrewder, a tenacious band who understood each other and worked together. There was no more quarreling. […]. Without Hazel, without Blackberry, Buckthorn and Pipkin—Bigwig would have died. […] There was no more questioning of Bigwig's strength, Fiver's insight, Blackberry's wits or Hazel's authority. (18.3)
This is a good lesson for all of them, from Hazel to Acorn and Speedwell. But it's an especially useful lesson for Bigwig who transforms from a somewhat selfish fighter, into a rabbity superhero, who fights to help people and listens to Hazel's instructions. This is the change in character that gives chapter 46 its title: "Bigwig Stands His Ground" and gives him that great line: "My Chief Rabbit has told me to defend this run and until he says otherwise I shall stay here" (47.55). It's hard to imagine Bigwig saying that at the beginning of the book, since, as Hazel thinks to himself, Bigwig won't like to take orders. But Bigwig shows some character growth, from a rabbit who was only looking out for himself, to someone willing to fight and die to protect the other rabbits.