I agree that the quest is bad design since it forces you to use a weapon. Not everyone likes the weapon so its unfair in a way. BUT, and here is the kicker, it says you need to have it EQUIPTED. Not saying you need to get kills with that weapon. You can use your special, heavy, grenades, melee and super to get kills. If you are up against good players, just unequip TLW and it wont effect your quest. There are also other ways to complete this part. SRL for example (if it comes back). Just have it equipted, kill some adds on the track and try to stay alive so your K/D (its crucible after all) stays positive. You complete this part of the quest in like 20 races. Boring but it works. Good luck.