I think he/she didn't kill her outside when she arrived because they wanted to terrorize her. This almost seems like something from a horror movie where the killer is sort of toying with the victim. I think this was extremely personal to this person and just shooting her wouldn't satisfy them. This would explain the complex disguise and unconventional murder weapon. When I first watched the video, I was instantly reminded of the killer from the movie 'The Prowler'. It's just extremely creepy. I really believe this person had a dark and sinister intention of doing more than just a one and done murder.
The whole setup was certainly consistent with terrorizing Missy, if that's what they wanted to do. We don't know what sort of confrontation and interaction occurred, and there's plenty of room for it to have been something that took a bit of time and included something personal.
But, the setup doesn't really tell us that's what they MUST HAVE had in mind. It could also mean they wanted to do a robbery and not be caught, and they took burglar tools, as well as a gun in case they were caught in the act and needed it to get away - and Missy walked in on them, and was killed as a result.