How should one act regarding a father's request for a sum of money to invest, given a previous experience where the money was not recovered, the son's desire to secure his money, and the fear of incurring the father's anger?

As long as your father is well-off and does not need your money, you are not obligated to give him what he wants, nor is he lawfully entitled to take anything from your money, according to the majority of scholars, as mentioned by Ibn Qudamah. He is only permitted to take from his child's wealth out of necessity, not unconditionally.

You are not obligated to give him your money for him to trade with it for you if you fear for your money.

However, if giving him some money as a gift or for mudarabah (profit-sharing) does not cause you harm, then do so to appease him and avoid his anger, as Imam Malik ruled in a similar situation.

It is important to note that mudarabah has specific Sharia conditions, including: the profit share must be known, the capital must not be guaranteed, and any loss must be borne solely by the capital provider.

If giving him money would cause you harm, apologize to him gently and politely, and use wisdom and diplomacy to mitigate his anger. Be keen on honoring him and showing kindness, while having sincere intention for the sake of Allah and seeking reward. For honoring parents is among the best deeds, and patience with them is a cause for Allah's pleasure.