First off, I won't make any attempt to match the beings of Tolkien's work to Christianity, as he said nothing in his work was intended to be an allegory.
Also, I don't think that Loki could fit in anywhere in Tolkien's Cosmogony. He is indeed a trickster character: this means that it was not always clear whether he was good or bad at heart (although that explanation is a little lacking). Thus, he certainly doesn't fit in as Melkor.
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"He was sent by a mere prudent plan of the angelic Valar or governors; but Authority had taken up this plan and enlarged it, at the moment of its failure."
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