Am I obliged to pray in the mosque or at home, taking into account that my mother is ill and cannot move except to use the restroom, and I am her sole caregiver?

If your mother always needs you close by to serve her and take care of her affairs, then you are excused from attending congregational prayers, as stated by scholars. This is similar to nursing a close relative, even if there is no fear of death or loss for them. However, if she is able to manage her affairs independently during your absence, then you must attend congregational prayers.