Could the anthropic principle taken into conjunction with our continued existence be used to show that the MWI is more probable? Here's the situation that I'm thinking about: it's well known that we've come close to nuclear annihilation multiple times, either by accident or by tensions between the US and the USSR in the Cold War. If the MWI is correct, it is probable that a good chunk of these worlds have been reduce…
To my understanding, the anthropic principle is basically: among the near-infinite set of possible physical laws, we can immediately rule out the ones that would prevent us from being around trying to figure out which physical laws govern the universe.
I think there's two varieties of this. The "weak anthropic principle" says, that doesn't mean those laws couldn't exist, there's no weird metaphysics around it, it's just that if the underlying physical laws didn't allow us to exist, you'd just have a universe that didn't have anyone in it and so the physics of that universe would be a moot point that didn't matter to anyone.
The "strong anthropic principle" says, wait, no, there is some metaphysical significance to this, and it's somehow not even logically possible for a non-anthropic universe to exist. I find that a little self-absorbed on the part of humanity as a whole, but that would tie in a little better to your quantum suicide idea. The weak anthropic principle, in contrast, would just kind of say, "yeah, there's a chance we all blow ourselves up, and if we do, I guess nobody will be around to care about physics anymore".
Even the strong anthropic principle doesn't prove much here. "Nuclear annihilation" probably wouldn't actually result in complete human extinction. Some people would survive. Individual quantum immortality doesn't stop you from shooting a hole through your brain and being neurologically disabled for the rest of your life, so collective quantum immortality also wouldn't stop you from living in a post-apocalyptic hellhole. Also, even the strong anthropic principle doesn't actually mean that humanity can't go extinct; after all, it didn't prevent the universe from existing for billions of years before humans started to exist. Maybe the real strong anthropic principle doesn't even apply to us, it applies to a species of avian hive-minded critters on a planet circling an unnamed star in the Andromeda galaxy, and as a weird side effect of living in a universe that is metaphysically required to permit for the existence of the Andromeda flying hivemind, humanity kind of shows up for a few hundred millennia and gets to do some physics before getting wiped out by a swarm of paperclip maximizers.