Is it permissible to give the father's bequest, which stipulated the expenditure of one-third of his wealth for charitable causes, to his poor son to pay off his debts and assist him, knowing that the sons are authorized to determine the recipient?

The will cannot be disbursed to the poor son, because the executor is merely an agent in distributing the money, and he is not permitted to give the money to his own inheritors among his relatives, nor to the inheritors of the testator. Moreover, merely giving to the poor does not constitute an ongoing charity (Sadaqah (definition) Jariyah). What was mentioned in the question, that the man gave his inheritors the choice, refers only to what they deem to be an ongoing charity.