Do you have an activision account? I think you need one to play online. If you play "online" in a private match the trophy should pop when reaching round 16. But I don't know if you need PS Plus.
You don't have to kill Megaton. You'll reach round 16 even if he's alive. I had the same problem. If you want to play solo you have to play a private match. Don't set the game in main menu to local by pressing square. Set up a private match by pressing R1 in level select screen in zombie mode. Reaching round 16 in private worked fine for me and the trophy popped.