What is Biddy’s reaction to Pip’s request and what does this tell us about her character?

Again I think you are on chapter 19 here. Relations between he and Biddy and Joe do not improve, however, especially when he asks Biddy if she would try and educate Joe so that he could bring him up to another social level once the full extent of Pip's sponsor's fortune is given to him. Biddy brusquely tells Pip that Joe has no need, and does not want, to be brought up to another social level. Pip is treating the people he meets differently as well, especially Joe and Biddy. Biddy's reaction shows her to be level headed and articulate even in the face of being offended.