You don't have to tolerate hate speech. You can argue against it or you can go somewhere it can no longer reach you.
You aren't required to engage with speech of any form, hate or otherwise.
OP may feel he has a moral or personal obligation to speak up against certain kinds of hate speech. Kudos to OP, he is doing a valuable service. However, that's his choice and it's not mandated by any entity and he can opt out at any time.
You aren't required to enter a debate with hate speech. You aren't required to accept hate speech. You aren't required to believe it.
Hate speech is protected because it is speech, and because defining sub-types of speech into categories like "hate speech" cannot in any way be imagined as objective. And if we start only protecting subjective sub-categories of speech we are in for a much bigger world of trouble than having to hear the college religious nut go off about abortions again.
War - this is honestly more directly aimed at you, tbh.
You don't have to engage in free speech or provide responses to hate speech. Some would say that not reacting, that ignoring hate speech, denying it an audience, is the best thing that you can do.
I don't agree. I think it is vital to speak up against perceived wrongs; we all have our own set, but for instance, say something blatantly racist or sexist in my eyesight and I'm going to call you out on it.
Except you're totally right. It's exhausting. Sometimes you want to just have a conversation. Sometimes you just want to vent about your day. Sometimes you want to just feel calm, not incited to anger, not riled up. Sometimes you want to just walk away. The more standing up against hate speech you do the more exhausted you get.
But you are standing up because you choose to stand up because you see an injustice and refuse to let it stand. To me that is real character. It is a person with no skin on the line other than being a witness who says, "You know what? No. I will not be a witness to this, I will not let this stand." You are standing up because you know it sucks but someone has to do it, you know someone has to do it, and you aren't going to sit on your ass looking around for "someone." You'd sit on your ass forever if you did that.
Having character, having things that you as a person stand for, and then backing those morals and beliefs up with frequent refusals to tolerate those who propagate ideas and lifestyles which are completely counter to those morals? That is exhausting. Especially if you are a decent person who sees nuance in things and doesn't want to fight people all the time. Who wants to like people, give them the benefit of the doubt, who doesn't want to be seen as aggressive or antagonistic or "that fucking annoying fucker, bringing up race arguments again." (I imagine the Westboro Baptists do not find themselves particularly exhausted at the end of the day; I imagine they cackle with glee, actually.)
It is exhausting to stand up for what you believe in. But the only reason you are able to do so is because of free speech.
We cannot limit speech based on our feelings about the content because what will end up happening is that others will limit our speech because of our feelings and our content. We can only choose whether or not to tolerate it; to serve as an audience; to speak up, debate, or challenge it.
There are users here who have left because they got tired of standing up. More's the pity: no one has stepped in to fill their shoes. And that's what happens when voices of dissent get beaten down into silence, they stepped all over, forced to repeat themselves and their arguments again and again, forced to reduce their existence to constantly arguing with different bigots over the same shit every day because guess what? There will always be more bigots. There will always be more innocent ignoramuses.
It fuckin' sucks.
All I can say is shoulder on and be the voice of dissent and keep being it. Because the voices of dissent are vital, and few, and necessary.