The Aunt would have been Mrs Leigh Perrot,who lived sometiems in Bath with her husband Mr Leigh Perrot.

I quote from The Family Record book:

Mr Leigh-Perrot was an intelligent man with an enquiring mind,and his abilities would have stood him in good stead had he been obliged to adopt any profession; he was also of a kind and affectionate disposition,combining an easy temper with ready wit and much resolution of character.His wife did not have the same amiability of nature,but was nevertheless highly respected: she was very shrewd where financial matters were concerned,had a great idea of the claims of family ties and akeen sense of justice as between herself and others

My reading is that if JA and CAasndra were to go to Bath,they woudl end upacting as unpaid copaions ,running errands etc.Her aunt was making what seemed, on the face like a kind offer,but in reality shereally wanted to profit from it.