We gotta give Barton props for being earnest. He takes his job very seriously, as Lina finds out when she brings a message to the mayor on her first day working as a messenger. "He was a big man, with wide shoulders, brawny arms, and a thick neck. But his head looked as if it didn't belong to his body—it was small and round and topped with a fuzz of extremely short hair" (2.63). Obviously being a guard is about more than physical strength, but it sounds like he's got that area covered and then some.
He's very efficient in having Lina wait in the Reception Room for the mayor. Unfortunately, Lina doesn't stay put, which gets Barton in trouble. The mayor reprimands him for not watching her more carefully. It doesn't even seem to occur to Barton that the mayor might have something to hide. He's far too earnest for that.
When Lina and Doon show up to snitch on Mayor Cole for stealing, they tell Barton, hoping that he'll pass along the message to someone who can do something about it. At the mention of an important and secret message, "His face brightened." (12.64) Clearly he's been aching to have this kind of sense of purpose. When the kids leave, they see Barton "standing up, leaning forward, his eyes shining with important news" (12.74).
It's nice that Barton wants to do something to help the city, but he picks the wrong people to confide in, since we never see him again. Probably he's just been thrown into the Prison Room by the guards who are the mayor's accomplices, but still. Poor Barton—he just wanted to help.