The narrator of Poe's story asks the same question:
“But what purpose had you,” I asked, “in replacing the letter by a facsimile? Would it not have been better, at the first visit, to have seized it openly, and departed?”
Dupin's answer is that if he had done as the narrator suggests and openly takes the letter, he would have put himself in danger. By leaving a copy of the letter, he has an advantage over the suspect, Minister D---. The suspect has no idea that anyone besides himself and the queen know what is written in the letter. He believes he can continue controlling her through blackmail. Dupin says that Minister D--- will trap himself:
...being unaware that the letter is not in his possession, he will proceed with his exactions as if it was. Thus will he inevitably commit himself, at once, to his political destruction.
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This part of Poe's story seems weak. If Minister D-- is so terribly clever, he certainly ought to be suspicious when Dupin visits him, not once but twice, for no apparent specific reason but just ostensibly to have a casual conversation. The Minister ought to have some notion that Dupin has had dealings with the Parisian police and even that he has assisted the Prefect of Police. On his first visit to the Minister which is supposed to be "quite by accident," Dupin is wearing "a pair of green spectacles." Surely that would make Dupin look weird and give the Minister cause to wonder what his visitor is up to, especially since the blackmailer is well aware that he is under heavy surveillance and that his rooms have been meticulously searched at least twice. It was odd enough for Dupin to pay the first visit unannounced and for no apparent purpose, but when he came back the second time that should have been sufficient to make the Minister more than merely suspicious but positive that Dupin was after the purloined letter.
At the conclusion of the story Dupin tells his friend:
D--, at Vienna once, did me an evil turn, which I told him, quite good-humoredly, that I should remember.
The Minister should remember (1) that he did Dupin an evil turn, and (2) that Dupin promised that he would get back at him someday. These are additional reasons why the Minister should be on guard.
It is hard enough to believe that the police would not find a letter hidden in plain sight, but Dupin's two visits to the Minister make the story even harder to believe. "The Purloined Letter" does not measure up to Poe's "The Gold-Bug" or "The Murders in the Rue Morgue," both of which are more logical and more dramatic.
The real question should not be "Why does Dupin replace the real letter with a facsimile?" but "Why does Edgar Allan Poe have his character Dupin replace the real letter with a facsimile?" Dupin gives his friend a complicated answer consisting of several different reasons. But Poe was concerned about the impact of his surprise solution. He wanted to make sure that no one but Dupin would know that the real letter had been recovered. If Minister D- realized that the letter had been taken, he might inform somebody else, and that person might inform yet another person. The letter was the focal point of political intrigue, and there must have been others involved. The news might have spread all over Paris. The cat would be out of the bag--and Dupin didn't know how long he would have to wait before he could turn the letter over to Prefect G- and collect that 50,000-franc reward.
If D- knew that the letter was gone, there is no telling what he might do. He is a clever and dangerous man. As long as he doeosn't know the hiding place has been discovered, he is pinned down. He will do nothing. Dupin doesn't have to worry about him--and neither does Poe.
The police were keeping a close watch on D-. They themselves might find out somehow that the letter was no longer in his possession. D- might even go to see the woman from whmo he had purloined the letter and tell her that he had destroyed it out of compassion or say something else that would reveal he no longer had his hold on her. If word got back to G- that D- had lost possession of the purloined letter, there would be no surprise in Poe's story and no reward for Dupin.