What you explained is also Aspie=ish. What kind of help is this child getting to help her learn how to interact appropriately with other people? Aspies need help. They don't pick things up by observation.
They tend to have specific interests, very narrow, and talk about them nonstop. They may like to be around people, but don't know to have a two way conversation. They tend to monologue or get bored. I am not going to try to push it beyond this post, but I think you are seeing a behavior problem where a neurological problem exists. Please don't ever call her a brat or assume she is trying to be "bad." My own mother did this...I never forgot it and it went on forever and just got worst. The best thing in my opinion that you can do for both of you is to get her a neuropsychological evaluation rather than guessing or maybe listening to teachers (who do not diagnose disorders) or relatives or neighbors or even a pediatrician (this is not a pediatrician's field). A neuropsychologist is a psychologist with special extra training in the brain and they test extensively in all areas. They are NOT the same as neurologists. You can find them at teaching hospitals and children's hospitals.
She is not a brat. Something is going on that causes her not to understand how to act appropriately with other people. There is no one way an Aspie behaves. At any rate, if you don't get her a neuropsychologist test, you are not helping her; you are condemning her without having her completely evaluated to see what is really going on. You sound very hostile toward just a little girl, no offense, but she IS just a little girl. You are the adult. It is your responsibility as her parent to take her for the best evaluation then help/interventions based on that evaluation. That is all of our jobs...to help our children, not to think they are brats, a word that she WILL remember if you ever use it in front of her.
Let the professionals handle it and maybe get therapy for yourself too so you can deal better with your daughter while she is being diagnosed and helped. If you don't like a neuropsychologist (I feel they are the best diagnosticians the U.S. offers), then at least see a psychiatrist (the one with the MD). Do not keep on blaming your daughter for behaviors that are probably not her fault. The longer she gets no help, yes, the worst it will get. If there is a father, involved him too. If you are alone with her, I know it is stressful and hard, but she is only a fourth grader. Try to make things better for her and don't call her names. That will not end well. Do not compare her to or expect her to be like neurotypial fourth grade girls. This is setting her up for failure. She can not be like them. She is herself and she is fine the way she is and she can even get better, but she is not going to be just like her peers are.She needs your support, not your anger and wishful thinking that she be like the other kids. Stop talking to other mothers. Talk to those only who "get it." Read. Learn all you can to help your daughter and yourself.
A ten year old can't have borderline personality disorder. Stop reading the internet. Go to the REAL professionals. Find out what is REALLY wrong. Autism Spectrum Disorders (ASD) makes it hard to be flexible and many have fluctuating emotions. They are not the same as borderline.
Hugs and good luck.