In any case, the point here seems to be that someone is adopting a pose, that is to say, he (or she) is standing in a certain way with a certain expression. He wants to appear very tragic, that is to say, sad or pitiful or deeply upset or the like.
"Spare us" means that the speaker is asking him to not act in this way. You might also say, "Spare us the sour look on your face" or "Spare us your crazy ideas about who will win the game in Koshien Stadium this weekend."
The biggest problem is the word "existential." Writers such as Camus and Sartre were existentialist. What were they saying, these authors? Well, I have never been sure, but many existentialists were thought to wear black and always be lamenting the deeply tragic character of life. I'm sure it is deeper than that, but I suspect that that is the kind of interpretation of existential that the speaker you have in mind is thinking of when he speaks these words.
I'm sorry for the excessively long discussion of this sentence. I guess it could be interpreted as, "Stop standing there looking like your mother just died. We don't want to see it."
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