My answer may make it sound as though I am not taking your concern seriously, but please don’t think that. That answer is: ask her. Take her aside and talk with her individually; don’t let her sister stand nearby and kibitz. At first she is likely to say, “I don’t know.” And I’d stop the conversation there with the request, “Well, think about it.” Then, from time to time, come back to it with her. If she suggests anything, no matter how unlikely, agree and say, “I can see how that would bother you.” Then ask her to think about what might be done to correct the situation. If she can’t ever come up with anything, help her. “I think maybe you feel the other children don’t like you.” Also, at some point, ask her if she would prefer to stay home. She will probably say “No.” Keep the dialogue going as a sort of secret between you until you either have it worked out or you are convinced it isn’t helping.
Frankly, I don’t see any need for therapy at this point. But I’m a conservative on the issue of sending children for therapy when they have only minor adjustment problems. You report that everyone recognizes the crying at school is fake; she has been invited to visit some of the friends made at school; and she loves the projects and things she is learning. So that’s not all bad. Schedule an appointment with her teacher and discuss it honestly with her. And make certain you are not contributing to the problem by secretly enjoying the fact that she would rather stay home with you.
Our parenting advice is given as suggestions only. We recommend you also consult your healthcare provider, and urge you to contact them immediately if your question is urgent or about a medical condition.