Why do you think Jamie objected to Claudia covering his mouth the way she did?
In Chapter Seven, after renting a post office box, Claudia and Jamie return to the museum.
Both of them badly want to deliver their letter to the museum officials, but they're afraid of the consequences if they do. So, they decide that they will select a messenger from one of the school groups that regularly come through the museum.
They look for a school group in all the usual places: the Egyptian Art area, the Costume Institute, and the Arms and Armor section. Soon, they settle on a school group filing out of the Egyptian wing, past the entrance. However, as Jamie and Claudia listen to the conversation of the schoolchildren, they come to the realization that these are Jamie's classmates.
Meanwhile, Jamie's eyes meet Claudia's, and he opens his mouth as if to speak. Without a moment to lose, Claudia clamps her hand over Jamie's mouth to prevent him from speaking. Essentially, she wants to stop Jamie from exposing both of them. For his part, Jamie is upset that Claudia doesn't trust him. He feels that he has a little more sense than to expose both of them to his classmates. Jamie thinks that Claudia should have trusted in his good sense, so he's a little offended when she doesn't do so.
Still smarting from what he thinks is her high-handed action, Jamie threatens to re-join his school group, but Claudia has a better idea. She tells Jamie that he should capitalize on being in the third grade and pretend that he's part of the school group from Greenwich (which in theory, he actually is). Then, he can deliver the letter himself at the museum office, claiming that he's doing it on behalf of someone who belongs to the third grade school group. In the end, this is exactly what Jamie does.