How can one balance his desire to bring his wife to live with him and caring for his sick mother, who relies on his wife for her care, especially since the mother refuses to move in with them?

There is no harm in bringing one's wife over if the questioner's situation is as described. If the siblings are unable to care for the mother, then someone must be brought in to care for her. The questioner should apologize to his mother and explain his need to bring his wife over, emphasizing his desire to bring her along with his mother, while continuing to show her filial piety.