This thought from Rabbi Shulman’s brother is so perfect, I’m just going to pretty much copy it straight. Unfortunately, I do not have the הגדה ישמח אב, so I will simply copy Gil Student’s summary:
We have an odd custom before הלל:
Why open the door? Why this diatribe about הַגּוֹיִם אֲשֶׁר לֹא יְדָעוּךָ? It’s very politically incorrect, but that is mostly an optical illusion. We assume the word גוֹיִם is an insult, from the Yiddish, but it’s really not. The diatribe is specifically about the destruction of the בית המקדש:
But that only makes the question sharper: why this? Why now? It’s the seder, not Tisha B’Av!
To answer this, we have to look at the seder as described in the Mishna:
Why do we split הלל up like this, part before ברכת המזון and part after? הלל on the seder night is different from other times we say הלל:
The fact that this הלל is a שירה means that it is supposed to be the spontaneous outpouring of emotion. חייב אדם לראות את עצמו כאילו הוא יצא ממצריים; לפיכך אנחנו חייבין…[ל]אמר לפניו הללוי־ה. It doesn’t have the limitations that the ritualized קריאת ההלל has. So we’re allowed to split it up. But why do it? Why not say all of it now?
So that nicely explains the custom of opening the doors. But what about שְׁפֹךְ חֲמָתְךָ?
This is when we sing הלל של שירה. If we are truly in the moment, we should feel it. We open our doors to join with the rest of בני ישראל in ירושלים, but then we look out and see…St. Louis. We’ve even poured the fifth cup, for the fifth לשון גאולה:
We invoke אליהו, the harbinger of גאולה:
But it does no good. Our anticipation of singing הלל is dashed.
So we break out with this antithesis of הלל, a song not of thanksgiving but of vengeance. It only lasts for four psukim and we get back under control and realize we still need to show הכרת הטוב to ה׳ (that’s the meaning of דיינו), and finish our הלל.
שְׁפֹךְ חֲמָתְךָ reminds us that we do not have a real seder. A real seder is playacting the night of יציאת מצרים, with a קרבן פסח in ירושלים. We are only playacting at the act of playacting, זכר למקדש, and for a brief moment our emotion leads us to an anti-shirah. It reminds us of how much we are missing and how much we await ביאת המשיח במהרה בימינו.