Atticus does not want Scout to tell Miss Caroline about their compromise because Miss Caroline does not want Scout to read, and Atticus does not want to antagonize her.
Scout is really looking forward to going to school, but she doesn’t like it at all when she finally goes. Miss Caroline spanks her and makes her stand in a corner, for one thing. One of the reasons that Scout and Miss Caroline get off on the wrong foot is that Scout is so precocious. She knows how to read, and this is disturbing to Miss Caroline, because she does not know what to do.
Miss Caroline told me to tell my father not to teach me any more, it would interfere with my reading.
"Teach me?" I said in surprise. "He hasn't taught me anything, Miss Caroline….” (ch 2)
When Scout goes home and tells her father she does not want to go to school any more, he realizes what happened. He tells her that there is such a thing as compromise, and this is a good situation for it. She has to go to school, but they can continue reading.
When Scout asks why she can't say anything at school, Atticus responds in formal language.
"I'm afraid our activities would be received with considerable
disapprobation by the more learned authorities." (ch 2)
Basically, he means that Miss Caroline would not approve. If she finds out that Atticus is still going to read with Scout, she will get upset since she considers it teaching her to read, which is not appropriate. Miss Caroline is a new teacher and simply does not know what to do with a kid like Scout. Atticus uses the moment as a chance to teach Scout about empathy.