Given that it doesn’t have to be a self-sustaining base on Mars, I think the full resources of the earth could establish a refuge there to wait out the cosmic hiccup. Especially since we could start immediately and fail a few times. As has been shown by many examples-the latest is SpaceX, if you want to get something done, be prepared to fail a few times. That lack is why NASA can’t get anything done. With that constraint removed, Mars wouldn’t be impossible. Yeah, we would be concerned about radiation exposure for the crew etc, but not enough to not go.
Now if Ceres happened to be the one asteroid that would be protected-heck with Mars. Take the easy path. An entire (dwarf) planet made of ice, low gravity to make landing and more importantly leaving easy, almost no atmosphere to interfere with space operations, what’s not to like?
Gather the sperm and ova of millions of people and freeze them. Similarly gather the DNA material of every conceivable plant and animal, excluding bitey insects.
Gather a crew of healthy women astronauts of child bearing age.
Calculate how big a ship you would need to carry this stuff, how long it would take to send it to the other side of the sun, and back, and how much fuel would be needed.
If you can leave stuff on Earth that could survive this “event” (for instance, if it was like being in an autoclave for 24 hours) then you can leave facilities on Earth to start repopulating the planet once the DNA material gets to it. Otherwise, you need to expand the ship to carry such a facility.
Start a program to see which animal species can serve as hosts for the fertilized eggs of other species. If you’re lucky, maybe you’ll find single species that can be implanted with and give birth to dozens or hundreds of other species. 'Cause you’ll need some kind of Ark, but you can’t possibly carry a breeding pair of every species.
In the remaining time, put a lot of effort into developing artificial wombs, and if you’re lucky again, you can maybe dispense with the Ark idea.
Build and launch your ship in time to get it into shelter. It should be able to return in short order without landing anywhere.
Then trust your intrepid women astronauts to repopulate the world!
I hate to tell you this but I don’t see how you can cool the expected 200 C to 250 C temperatures at 11 km down. You need a source of cool air and routing it through 11 km of rock (Which is heated already!) is not going to work