Is it permissible to pressure the blameless party and deem them at fault—for the sake of reconciliation between people—justifying this by saying that we are in a reconciliation, not a judiciary, and does this have a basis in Islamic law, or is it considered an injustice?

Firstly, pressuring the wronged person to admit fault and convince them to reconcile is forbidden, because it is an unjust reconciliation and contrary to the law of Allah, which commands justice. The wronged person must be enabled to take their right, and then they may be asked to concede a part of it willingly, not under duress.

Secondly, the judge should not invite litigants to reconcile unless the truth is obscure to him. However, if it is clear to him, then it is obligatory for him to rule with justice. The judge may invite them to reconcile if both parties agree or if he sees a preponderant benefit, such as fear of the dispute escalating, and the reconciliation must be just.