Bonus Fact: And although Core!Frisk doesn't have her own Omega Timeline, she still helps humans and monsters reach the local Omega Timeline.

You could say, Core!Frisk is Bear!Error's only friend. And that's why this interpretation of Core!Frisk doesn't have her own Omega Timeline, as she is busy trailing this bear through universes and multiverses to prevent any problems he might cause.

What's more, he will only leave your AU, or any other, at her behest, because the inhabitants of the AU would suffer more than they could manage such a powerful adversary.

To avoid this bear-like hibernation in your AU, there exists an alternative version of Core!Frisk, who is the local caretaker (a zoologist or bear herder). She is the only one towards whom Bear!Error has a more positive disposition and will never attack.

While his direct presence in a particular AU gradually destroys the world itself, he can sleep until the very end, until the world perishes.

He particularly loves to arrange unexpected "hibernations" (especially during a season lucky for it), and thus he can find a place in your universe and fall asleep.

That's right, unlike Error, Bear!Error doesn't aim to destroy all universes. As a bear, this is "uninteresting" to him (fortunately). He lives more for the sake of living: he travels through different universes and multiverses in search of good places to sleep and eat.

Although, deep down, due to his nature, he still feels disgust towards errors and anomalies, but he doesn't rush to destroy AUs, since he doesn't really... do that?

Bear!Error's personality doesn't change much from the canonical Error, except for the aforementioned bear-like mannerisms and a more patient and restrained temperament.

If we're talking about Bear!Error – this bear is sentient and can speak. However, he mostly prefers to behave like a bear and often acts like a bear, because he finds it more convenient. After all, what sane person would approach a bear if you don't anthropomorphize it... right?