It is in fact more logical to pit Huan the dog against an evil cat, and here in the Reading Room, we're always chasing after logical answers and extrapolations as desirable things, but it makes for a far inferior story, in my opinion. So, logical connections may serve as a beginning in a story, but they needn't be adhered to. I love Huan fighting Sauron in the Silmarillion tale--one of my favorite parts. Who cares about a dog/cat fight? What's more important is that Huan is bound by loyalty and friendship to Luthien, so he does the right thing and fights the bad guy.

Long Felagund watched them, and love for them stirred in his heart; but he remained hidden in the trees until they had all fallen asleep. Then he went among the sleeping people, and sat beside their dying fire where none kept watch; and he took up a rude harp which Bëor had laid aside, and he played music upon it such as the ears of Men had not heard; for they had as yet no teachers in the art, save only the Dark Elves in the wild lands.

Now men awoke and listened to Felagund as he harped and sang, and each thought that he was in some fair dream, until he saw that his fellows were awake also beside him; but they did not speak or stir while Felagund still played, because of the beauty of the music and the wonder of the song. Wisdom was in the words of the Elven-king, and the hearts grew wiser that hearkened to him; for the things of which he sang, of the making of Arda, and the bliss of Aman beyond the shadows of the Sea, came as clear visions before their eyes, and his Elvish speech was interpreted in each mind according to its measure.

Thus it was that Men called King Felagund, whom they first met of all the Eldar, Nóm, that is Wisdom, in the language of that people, and after him they named his folk Nómin, the Wise.

1. I don't know if this is the version that appears in "Lost Tales" or "History of M.E." In this version, Tunuviel's father is Tinwelint, and he's kind of a jerk.