You have to have TLW equipped but you don't have to get the kills with it.
The issue is you get 1 point for a kill, 2 for a headshot and lose 2 for a death so you have to be pretty good at crucible to get it done.
FWIW the last stage of the quest is an harder version of the Shield Brothers quest thats 290 light, Nightfall level basically, so it's a pretty tough quest overall