This argument is lots of fun! Thanks for offering it. Whether it works, of course, may be another matter. There are many places to try to poke holes, and others on the panel may have their needles poised over other places. But here's my pick.

Your first premise says: if the world's existence is contingent, then there is a possible world where the world doesn't exist. That sounds a little odd, and the oddness suggests that we need to be careful. Most of us think at least this much: the matter that surround us didn't have to exist. When we say that the world's existence is contingent, it's plausible that we mean something like: "The matter that happens to exist need not have existed." That seems plausibly true; it's certainly not obviously false. Consider: if physics has it right, the matter of this world obeys quantum theory. But other laws of nature seem at least possible. Arguably, matter that doesn't fit quantum principles couldn't be this matter -- the matter that makes up the stuff of this world. And so it seems possible that this world might not have existed. In that case, however, the argument that the world is non-contingent doesn't get off the ground.

More generally, philosophers disagree about just how robust an interpretation we should accord to possible worlds talk. If talking about possible world lands us immediately in highly counter-intuitive metaphysical claims, that would give us reason to wonder if possible worlds talk was the right way to go in the first place.

My view, as you can no doubt see, is that the challenge here is to say just where the argument goes wrong. (This isn't because I want to defend theism, by the way, though I have no objection if someone else wants to.) But that's not meant as an insult to the argument. On the contrary, I think there's a lot to be learned from pondering it.