Is one absolved of responsibility before Allah by informing one's brother about exchanging a gift, and the doer's regret, while the brother refuses to take back or claim the gift?

You have fallen into two prohibitions: betraying a trust and lying to your brother. You must repent to Allah Almighty and return the right to your brother. It is not a condition to inform him of what happened, for fear of a rift; rather, return it in a way that preserves affection. If you are unable to do so, then ask for his forgiveness.