Is it permissible for one sister to pay a sum of money to her mother for the purchase of a new appliance for another sister, given that the sister paying the money is content, but the sister benefiting from it is not convinced?

If you both agree that your sister will pay a sum for your device, or if she donates it and you agree to leave the device for her as a donation or in exchange for a price, then there is no harm in that. However, if you are not content with leaving the device for your sister in exchange for what your mother took from her, then you are not obliged to do so, and you must inform your mother of this.