ANSWER:
No, you must not go out in front of non mahrams like this. You must wear the loose abaya over these clothes and convince your mum that what she’s saying is not permissible in Islam. Be diplomatic but firm and show utmost respect and never raise your voice while dealing with these kind of situations with your parents. It is also not permissible to mix with non mahrams even if you are wearing abayah. Make this clear to them.