I really, really hope I'm wrong, but what if he incapacitated, drove her to a steep area/cliff side and just threw her down, leaving her to die alone. No bullet left to trace back to him, injuries "naturally sustained" in the fall that can't be blamed on him plus the added cruelty of letting her die slowly out in the elements. I suspect he wanted to actually physically kill her and did so either by strangling her or shooting her, but with the tranquilizer cap found, I'm not fully committed to that theory any longer.