Is it permissible for a working wife to bring in a housemaid at her own expense to manage household affairs, despite her husband's objection?

Allah has made the man the maintainer of his household. Therefore, it is not permissible for a woman to dispose of any matter related to it except with his permission. This is especially true in the presence of a maid, due to the potential legal prohibitions that may arise. This does not mean that a woman is not allowed to dispose of her own private wealth.