A special note--Beornings have a slightly reduced pool of LIs than other classes. I don't know the reasons why, but it might be due to their late introduction to game. This applies especially to Second Age and First Age weaponry that drops on the landscape or in instances, even though they can cosmetically use them. However, the crafted and bartered LIs remain unaffected. This just means you'll need to do some extra farming for certain ones.
As usual, I'll share what I've collected here. If you see a skin here that's obtainable elsewhere, please let me know in the comments.
War-axe of Khand - Bartered from the Defenders of Minas Tirith Quatermaster