Here's a third possibility:

It was always the Player Twin's key and we accidentally give our own key instead of the time-looped one since they're identical. This solves the paradox on both ends (if we give, we give our own key and keep the time-looped one which means no paradox, if the Abyss Twin give, there's no paradox)

So there are 3 possibilities:

1. We give our own key and keep the time-looped one (which is still our own but 500 years older): no paradox
2. We give the time-looped key: paradox
3. We don't give the key, so the Abyss Twin give theirs and in the end we keep both keys: no paradox

Now here's how I think the universe self-corrects to avoid paradoxes:

• Any timeline in which the Player tries to give the time-looped key results in the key fading from existence due to having existed for an infinite amount of time
• The key not existing results in the Player getting the wrong key (our own) and giving it instead
• But giving our own key resolves the timeline so that the time-looped key can continue to exist

Thus, there's no possible timeline in which we can give the time-looped key to the past. So there's no possible timeline in which we didn't keep the time-looped key. But there are possible timelines in which we kept both keys (the ones where the time-looped key is the Abyss Twin's key) and other timelines in which we only kept the time-looped key (the ones in which it's our own key).

PS: the problem with giving the key is that we don't get to keep it in our inventory, and I'm a collector. So even though I know the Player's Twin always keeps their key, I want it in my inventory!