A Theory and a Question.
First, the question.
If Sweat was involved with killing the girls, what do you think his motive was?
My theory is that he was contemplating killing his girlfriend already. He wanted to make sure he could pull the trigger, both literally and figuratively. I think he choose the girls just to see if he could do it and get away with it.
Then, I think he shot his girlfriend and then probably dissected her and set her on fire. I think they found bullets in with the effects of her body and they match, somewhat, with the ballistics from the Taylor/Skyla murders, but they really need the gun to prove their case in court. That is why I think they need the gun, to be sure that bullets match both cases.