This is what the Briskers call "A Famous Question."
The Gemara in Pesachim 5a and 28b brings opinions that Chametz is prohibited on the fourteenth of Nissan in the afternoon, even before Pesach begins that night. The Gemara derives this from two pesukim. Rava on 5a derives it from Shemos 34:25 לא תשחט על חמץ דם זבחי, and Reb Yehuda on 28b derives it from Devarim 16:3- ומנין לאוכל חמץ משש שעות ולמעלה שהוא בלא תעשה שנאמר לא תאכל עליו חמץ. The language of the Rambam (1 Ch'uM 8) is
Tosfos asks, who needs this passuk? We know that sacrificial service can only be done during the daytime from Vaykira 7:38, ביום צוותו את בני ישראל להקריב את קרבניהם, and so the blood service can only be done during the day. Once sunset has passed, it is no longer day, and you would have to wait until the next day to continue- which is impossible, because all items that have been sanctified for use on the altar are profaned with the break of dawn. So who needs to know that Sunset profanes the blood, when you won't be able to use it until the next day, at which point it is profaned, on the basis of a different passuk, at the break of dawn? Although there is an opinion that items that are kept on the roof of the Altar are protected from the profanation caused by Daybreak, what of the other opinion, that being on the Altar affords no protection at all?
I have several major problems with Reb Chaim's answer. (I know that the person who asked the question was not worried about either the Rambam or Rabbeinu Tam, because there is no reason on Earth to think they agree with each other on the derivation of the pre-pesach issur and they certainly don't agree on the shkiah issue. He was just asking a clever what-if question. But Reb Chaim's answer has to make sense. And Reb Chaim was not playing games, because he is quoted by the Brisker Rov and by Rav Shach in his Avi Ezri 1 Ma'Hak 4, who says he was very happy to have thought of Reb Chaim's answer before hearing it from the Brisker Rov.)
Problem 1. While Shechita is a sacrificial act, the purpose of shechita is to produce sacrificial blood. The Gemara in Chulin (33a) states that unlike the shechita of non-hekdesh animals,
Problem 2. The Rambam, like Tosfos' first answer, doesn't hold that Shkiah profanes the blood. He says that you can't do the service after the day ends, because sacrificial service must be done during the day, and since you have to wait until the next morning to continue, daybreak will then profane the blood. So techincally, there is no "Profanation of the sacrificial blood caused by Shki'ah." There is only a forced end of the service of the day. So how can Reb Chaim say that it is day for shechita despite a tangential profanation of the blood. The Rambam clearly stated that the reason the blood becomes useless is because the day's sacrificial work has come to an end, and that includes the shechita. It is no longer "the time of Shechita."
In thinking about this second problem, I realized that even according to Tosfos first answer/the Rambam, the blood does become profaned at the moment of Shkiah. Since at the moment of shkiah its fate was sealed, there is nothing you're going to able to do with it, then even though the cause of that problem is the inability to use it at night and the inevitable profanation of daybreak, this causes it to become immediately profaned under the rule of Mishmeres/Hesech HaDaas, that items of sacrificial service have to be held in preparation, in expectation, in anticipation of service. Here, their uselessness is immediate with shkiah, so they become profaned at the very moment that their uselessness becomes inevitable. But this doesn't help Reb Chaim. The blood is only useless because it's not day any more, and if it's not day for the blood, it's not day for Shechita either.
3. Why would anyone think that the erev Pesach prohibition of Chametz, which begins when you can shecht the Korban, ends before you've eaten the korban? To me, it is self evident that if the Torah prohibits chametz when you can begin the sacrificial service of the Pesach, then how much more so is chametz prohibited when the Korban is ready and waiting to be eaten. This is not a kal vachomer, an a fortiori. It is too obvious to need any rule of a fortiori. There is no logic on Earth to think that chametz is prohibited during the initial preparatory stages of a korban but permitted during the pendency or the denouement of the korban. So although the Rambam uses a passuk that refers to "the time of shechita," it is obvious that the same prohibition applies FROM "the time of the shechita" until the korban has been consumed (at which point, of course, the Pesach prohibition of Chametz has come into effect.)