Is it permissible to give one of the needy children from their father's money, while the father is afflicted with Alzheimer's, as a documented and written loan, with the presence of siblings who are guardians over the money, and without compromising the parents' expenses and needs?

If the father is ill and unaware of his surroundings, his financial transactions are not valid, and his case should be brought before the Shariah court to have him interdicted. Until then, you must safeguard his money. If the court rules that he is competent, then you should not dispose of his money except with his permission. If it rules that he is interdicted, then it is not permissible to dispose of his money except in ways that benefit him. Therefore, giving gifts from it without compensation or lending from it is not permissible, because lending is a donation, and it is only permissible from someone who has the right to donate. Consequently, you are not permitted to lend anything from your father's money, especially since there is no necessity or benefit for him in doing so.