This article gives a lot of background as to why the door is opened in the first place, and its possible connection to Eliyahu and Moshiach. It brings a bunch of explanations why the door is opened later rather than earlier, but most of them don't explain why specifically at shefoch chamasecha. Some excerpts of those that sort of do:
Bais Ha-Levi (19th century), Parshas Bo, p. 15 and the Chasam Sofer (18th century) in his notes to Shulchan Orach, 480
He answers that the Gemara in Pesachim (109 b) asks how can there be a halacha to drink four cups of wine if there is a danger to eat or drink things in pairs—which is known as zugos. The Gemara answers since it is Leil Shimurim, there is no danger. So the Bais Halevi says that we specifically open the door when the fourth cup is drunk to explain to the person who would ask why isn’t there a problem of zugos. We show him that it’s not a problem because it is Leil Shimurim as we open the door
Rabbi Yosef Zecharia Stern (19th century), Zecher Yehosef, p. 39
He says the Gemara in Pesachim says another way that there is no problem of zugos is if one opens the door to the street. So that is why we open the door specifically at this point in the seder.
Rabbi Shmuel Rozovski (20th century), Mikraei Kodesh (Harri), p. 548
Rabbi Shmuel Ruzuvsky suggested that the possible reason why the door is opened by Shefoch Chamascha is because when we used to eat the Korban Pesach the halacha is that one cannot take any of it out of the house so they used to lock the door. After bentching, they would go to the roof to say Hallel, so they opened the door.
I admit these more explain why around the fourth cup than specifically shefoch chamasecha, but I guess it's the first possible opportunity closest to that point.