Is it permissible for my sister to exploit my mother's commands to me to serve her, considering the harm and excess this causes, which may lead to her persistence in relying on others?

It is not forbidden for your sister to ask your mother for help so that your mother can order you to assist her. If you are bothered by this and see it as accustoming her to dependence and a lack of responsibility, then explain this to your mother respectfully or intercede with someone who can convince her. If your mother insists on ordering you to serve and you fear her anger, then obey her as long as no harm befalls you, for pleasing the mother and avoiding her anger is required due to her immense right.