Is it obligatory to preserve the stability of the home for the sake of the children and succumb to the wife's demands not to help the brothers; or are there other counsels?

A wife has no right to prevent her husband from financially assisting his siblings, so long as he is fulfilling his duties towards her and their children with kindness. The husband should explain to her the virtue of maintaining kinship ties, and that it is a cause for blessings in sustenance and lifespan, citing the Hadith (definition): "Whoever wishes that his sustenance be expanded and his life prolonged, let him maintain ties of kinship." The husband is advised to strive for reconciliation, avoid anything that incites discord, deal with his wife wisely, and perhaps conceal from her the matter of his assistance to his family if possible.