How should a bodhisattva respond to evil?

The Bodhisatta usually renounced in cases he could not help harmless. As for general on this topic, may good householder investigate: Wisdom over Justice and Justice vs. Skillfulness. As for householder, being oneself strongly corrup, still attached to sense-pleasure, as the Buddha told, one not free from sensual craving is actually incapable to help anybody. This wisdom even the worldling, the Bodhisatta, already had and therefore went for the place where help needs to start: leaving corruption by oneself.

May person doubts that good householder would really sacrifices any of sensuality himself to help himself and others, right? Help the rat or the snake? Just momentary preferences... They are all just around to help theirs...

A Bodhisattva only has one tool he brings to the table: an awareness of the true nature of the world. On occasion that's enough. People who commit 'bad' acts do so from within a thoroughly encompassing web of illusion in which the act appears necessary, or justified, or righteous, and in which the self is reified and magnified. Merely being exposed to the presence of someone who is aware of the illusory nature of that worldview is sometimes enough to gravitate a change; it's like adding ballast to a ship tossed in a storm.

What we do isn't half as important as bringing that open understanding into whatever we do. We don't want to add to the situation; we want to subtract from it.

Bodhisattvas don't save anyone. Bodhisattvas show people that they can save themselves. A lighthouse cannot keep a ship from wrecking itself on the rocks; all it can do is let it be known that no ship has to wreck itself on the rocks. Anything else it might do cannot interfere with that one purpose.

I think you are under a misapprehension about bodhicitta, as something you can either have or not have. It wells up in all beings, because it is a deep part of the nature of minds to recognise other minds, and as we wish for our own wellbeing, to wish for their wellbeing. It is an intrinsic and spontaneous quality, which only layers of bad experiences or being misled can cause us to ignore. Bodhicitta can be cultivated, practiced, by contemplating this wish for the benefit of all beings, and prioritising it as a motivation for action in life. To put it at the centre of our causes for action, is to walk the bodhisattva path.

The Buddhas teaching is very demanding:

"Monks, even if bandits were to savagely sever you, limb by limb, with a double-handled saw, even then, whoever of you harbors ill will at heart would not be upholding my Teaching" -Kakacupama Sutta

First of all, note this is about ill-will, not action and reaction, but how the bandits actions affect your mind. Forgiveness, doesn't mean no consequences. You don't have to help push the saw, you have to attend to your mind, and use it for the benefit of all beings.

Bear in mind also, this is directed at full-time spiritual practicioners, placing attaining awakening above all other priorities. Householders, lay-Buddhists, have other duties and priorities which the teachings are clear they should not shirk - consider how a farmer doing pest-control helps provide grain for monks to eat (but a skillful farmer needs less pest-control).

There is also the Buddhist idea of a 'wheel-turning monarch' or ruler or emperor: someone worldly who must manage an army and deliver justice, but does so in a way that creates opportunities for many more to become awakened. See for example the Cakkavatti sutta. Emperor Ashoka is surely such a one, the first ruler to unify India, although he was said to have given up war & converted to Buddhism after the particularly bloody and brutal military campaign involved. He did bot then abolish that military though, but sought to make violence less likely.

The Angulimalia Sutta concerns the conversion and awakening of a bandit that wore a necklace of human fingers and had murdered likely at least 200 people personally as a religious task (see the 'thuggee cult' for Indian cultural background on how this much earlier practice might have worked). Shown the true nature of things, Angulimalia ceased to cause suffering, and he was protected from families of his victims by the sangha, although he would still have to face the karma his actions had generated.

The koan of Nansen kills the cat generates a lot of controversy, which I think that linked commentary does a good job of picking through. An insightful point I've heard about the cat, is that they are very murderous of mice - yet humans very much need to not allow mice populations to grow unchecked (eg, mouse plagues, where mice also suffer). So a monk asks, should we keep a murdering cat? Nansen acts like the cat: You don't want this pest-cat? There, done. Happy? Having a cat is a choice like this, it's not a perfect choice, but we make a choice. The cat is not a monk. But like a farmer, like a monarch, it can be skillful.

In life we face dilemmas, we must decide who to help and who not. But don't make this a practice of just making endless agonising lists for and against why - that won't 'save the cat'. Put bodhicitta at your centre, know the taste of liberation, and the answer will come, while it can still make a difference, from paying attention to the qualities of this very moment. Make your mind into a diamond sword of wisdom, and you will know whether to kill or save the cat, for the benefit of all beings.

People don't set out to be evil. That word has special baggage because of it's importance in Christian theology, so it's worth thinking about how to interpret it, discussed here: The dark side of philosophy?. There is a saying "We are punished by our sins, not for our sins", that can help us understand evil-doers harm themselves, harm their own minds, and fundamental cure is in showing them the true nature of things, not telling them what to feel (though constraining their capacity to harm is likely a priority). That is recognising their Buddha nature, their intrinsic capacity to awaken to things as they are.