Bellief: "I am crucial to the success of the guard."

A player in my group has this, and I’m feeling a bit stumped on how to challenge it. Any suggestions?

Why does the character have this belief, how did he come to believe it, and can you challenge him by affecting those underlying currents?
Mind you, I’ve not played mouse guard, this is what I suggest from playing B.W.G. Try to get to the core of the beliefs, to help players get the most from them in game. There is an example of a character whose belief had changed from “One man can make a difference” to “Only by working together will we succeed” after he had spent most of an adventure helping his teammates. (I am paraphrasing as I do not have my book in hand at the moment).
Have you tried asking the player how he would like you to challenge this belief? What did he have in mind when he wrote it? (Again, I don’t know how things go in mouse guard, but in burning wheel things like that are pretty much kept out in the open)

More practically, you could:
Have Gwendolyn ask for volunteers for a mission. A regular old mission.
Then you could have Gwendolyn ask for volunteers for an incredibly hard or morally suspect mission (like destroying a brood of weasel babies).

While away on an extended mission, you could create the pressing need for a guardmouse to stay behind to fulfill the mission while the rest of the patrol moves on.
For example, you could lure the patrol into Dark Heather to discover an invasion force building up. Someone has to send word to Lockhaven while the patrol conducts sabotage on the force. Is it more crucial to stay or go?

Hope that helps,