To make a cosmetic override, right-click the weapon part and choose 'Complete' with 1000 sacred metal fragments in your inventory. You will receive no item, nor XP for this. Instead you will unlock a cosmetic override in the Avatar Customisation interface.
Fragments from the Battle of Lumbridge can not be used to make items in The Bird and the Beast. This was confirmed by Mod Moltare in a livestream who indicated that metal fragments will only affect the world event they are released with.