I always appreciate a thoughtful Reply.
Re. universality: Every human being can understand from anyone's own experience of life that people have no choice but to effect choices. Everything about real justice follows from that observation. Beliefs divide people; our shared experience of life is something we all have in common. That gives real justice a universality that no belief-based idea of justice can have.
Re. the need for mutuality: The onus is on each individual person to respect all others (at least to the minimum extent). For justice to be present all must be doing that.
Re. justice without 'morality': That is what real justice is, because it involves no belief. The concern of justice is how people treat one another. Heretofore all ideas of justice have been a part of some larger moral code. All moral codes are based on beliefs. Real justice makes justice its own 'thing', separate from any morality--though it might be consistent with any moral code.