Is the increase I took from the currency exchange rate without my uncle's knowledge unlawful for me, or must I give it to my uncle, knowing that he agreed to a lower price?

If the maternal uncle appoints his nephew to buy dollars, then a friend increases the price, the difference belongs to the uncle, not to the agent, unless the uncle permits him to take it. This is because the agent's role is merely to execute the transaction, as Urwah ibn al-Jad al-Bariqi did when the Prophet, peace and blessings be upon him, appointed him to buy a sheep for a dinar. He bought two sheep, sold one for a dinar, and returned to the Prophet with a dinar and a sheep.