Would resorting to the judiciary to sell the deceased husband's house at public auction to pay his debt to the questioner's brother, in the absence of official documents proving the debt and the heirs' non-recognition of it, be considered an injustice to the deceased husband's children? And what is the solution?

We do not know how a lawsuit can be filed without documents proving the debt. However, there is no harm for your brother in filing a lawsuit to claim his debt from the deceased. If the debt is proven and the judge rules to sell the deceased's house by auction to pay off the debt, then there is no injustice to the heirs, even if it is sold for less than its market value, especially since they can pay off the debt and keep the house.