Have my brother and I erred by secretly giving our uncle his share of the inheritance without our father's knowledge, and is that considered an injustice to our father or to ourselves?

What you have done is not an injustice to yourselves or to your father; rather, it is the essence of correctness and wisdom for mending relations and bringing brothers together. The father's stance in preventing his brother from inheriting is wrong, and he must prove his right with evidence. You have done well by donating your uncle's share to him for reconciliation.