The Lannisters used to have a Valyrian steel sword of their own named Brightroar. That sword was lost though.

It had always irked his father that none belonged to House Lannister. The old Kings of the Rock had owned such a weapon, but the greatsword Brightroar had been lost when the second King Tommen carried it back to Valyria on his fool’s quest. He had never returned; nor had Uncle Gery, the youngest and most reckless of his father’s brothers, who had gone seeking after the lost sword some eight years past.- ASOS - Tyrion IV

Tywin had been trying to get a replacement for many years with no luck.

Thrice at least Lord Tywin had offered to buy Valyrian longswords from impoverished lesser houses, but his advances had always been firmly rebuffed. The little lordlings would gladly part with their daughters should a Lannister come asking, but they cherished their old family swords. - ASOS - Tyrion IV

Acquiring Ice must have thrilled Tywin. Not just a sword to replace Brightroar, it was a two handed sword. It was large enough that, when melted down, it could supply enough Valyrian steel to make two swords. Now the Lannisters who had been without such a sword for so long would own *two *of them. Politically, Tywin likely considered the possesion of two Valyrian blades to be boon for his house. What other house has two such blades?

Morally… well, Tywin doesn’t really care about morals. He cares about appearance, and the appearance of power above all else. This is a man who publicly despised Tyrion’s whoring, but actually had a tunnel dug to a brothel so he could visit without being seen. This is also a man who killed every man woman, and child of house Reynes because they wouldn’t pay their debts to house Lannister and had the gall to rebel.

It was a symbol of power and added legitimacy to Starks by tradition. Remove the symbol, and you weaken whatever it stands for.

Also, Valyrian steel was pretty much impossible to make anymore since the secrets of refining the metal was lost with valyria, which made the sword pretty much priceless. Some master blacksmiths knew how to work it, but not how to make the steel. So it was logical to add a few of these priceless blades to the Lannister arsenal, and melting the original down would help remove the link to the original owners and legitimise the new ownership.

I suppose they could have not reforged it and just renamed it or something, but these weapons as rare and revered as they are, will have records of it being used and by whom - a paper trail.

Anyway thats what I think.