When Lucy fights Angel for the first time, Gemini turns into her and has duplicates of her keys. That makes no sense, because there's only supposed to be one of every key. They even say so in that fight?
If Gemini can make duplicate keys, could Lucy hypothetically summon the Spirit King with all of her keys and then just use the Gemini copies afterwards?
Gemini has the power to copy things. So since Gemini can copy the keys doesn't mean that lucy would be able to use them. Duplicating doesn't mean to copy. Copying something like on a printer isn't the real design. It's a "copy" of the real thing. Meaning that she wouldn't be able to use the copy, but she would be able to use the duplicate. Saying that Gemini only uses copies not duplicates. So lucy would not be able to use the fake keys.
The Gemini "fake" keys use the power of the actual keys. That means if Gemini summon Tauras then Lucy can't summon him for the time being since he is already "summoned" and she can't force close because she didn't summon him herself.
When Gemini had the key, they made a duplicate (a.k.a. a fake key) which works the same so Lucy cannot close the gate. But when they become their body again, they will lose the duplicate, so it is not a real key.