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Engagement, huzzah!

Huzzah! Forewarning, I'm gonna be building off of your napkin math, because napkin rocket science math is fun.

it requires companies like SpaceX (or their competition as they come online) to get the launch prices down.

Absolutely. Given the scale of such a project like this, the price per launch would absolutely go down over time (assuming no bullshitery on SpaceX/other corporate entity's part.) Though your original price point of $10m/launch is a bit off. The Falcon Heavy for instance, costs roughly $60-90m depending on payload and destination, and whether or not the rocket is recovered.

Thus a 1km2 solar sail would weigh only 50kg (of sail material). Add another 200kg for some tensile frame and some control electronics and you’re looking at something like a Starlink mass to get 1km2.

Another way to get an estimate is to compare to a recent, modern launch. The JWST is a good comparison, especially since it is in a similar orbit/distance/mission. The whole thing weighs 6,500kg, with 350kg of that being the RCS/reaction wheels/comms/electronics/frame/etc all wrapped up in the spacecraft bus.

So a completed frame can reasonably have a payload of 6,150kg for solar umbrella activities. If we put 1/3rd of that into the umbrella frame and the rest into the umbrella material, that's 4100kg for sail material, or 82km^2^. How you're gonna built an extendable frame that extends into a 9km x 9km sheet is a challenge, but maybe surmountable. This is a significantly bigger scale than the 1km^2^ sats you've proposed, but if the weight allocation works with JWST something similar should work here. The solar pressure will increase the fuel needed to keep a stable orbit, but nothing that our pre-designed launch platform can't handle.

So that would be 731 of these JWST scale sats that need to be put into L1 orbit. JWST was launched with the Ariane 5, which costs $150-200m/launch. That's significantly more that then $10m/launch, but getting all the way out to L1 with a 6,500kg payload is hard. I wasn't able to find a cost associated with the JWST itself, only the development cost of ~$8.8 billion. But I'm gonna assume that the construction of the satellite itself was in the millions, if not billions. If it is even a single billion for just one of these, that's almost a trillion dollars for this project as a whole.

All of that for only a 0.1% reduction in sunlight. Not sure how much we need, but it seems small.

Okay, other options: we put the solar sails in a very high earth orbit (above the comms satellites) – doable, but you’ll require many many more of them as they won’t site between the Earth and the Sun during most of their orbit.

I have an even dumber, even more harmful, version of this that is just as fun to explore. Go up to the moon, build a couple rail launchers, and start launching shit loads of moon regolith into a high orbit around the earth, somewhere between geostationary orbit and lunar orbit. Eventually Earth will have it's own set of rings. We only launch everything for one week of the month every month to ensure the inclination of the rings stays somewhat uniform.

The benefit of this being once the infrastructure to do this is put on the moon, this can essentially run for free forever. We just have to be mindful of avoiding Earth's rings as we travel outside of our system.

Ironically, raising our albedo might be a decent local option – just mandate white roofs everywhere. Just under 3% of our surface is urban and white roofs would also help with the urban heat island issue. You can probably paint 0.2% of the surface white. Not as good as blocking sunlight, but useful. The bad part is, solar panels are all dark, and moving to solar decreases our albedo. So maybe this will just offset changes in our average albedo due to solar panels.

I think this is an almost guaranteed partial solution that we will end up doing. If you've traveled around at all, you'll notice that hotter climates tend to use white roofs, seemingly automatically. Home owners will automatically do whatever suits them for the climate, no matter how the climate changes. The problem here is so much of our buildings and infrastructure isn't roofs. So much of it is roads and parking. That's a lot harder to change the albedo.

A light nuclear winter sounds like a disaster – what do we do, nuke a few volcanoes to set them off prematurely? That doesn’t sound sustainable. Burn all the forests to release ash? Nope, that’s our carbon sink that’s burning…

Your last option reminds me of: Kill all the poor!

Yeah, these are the dumbest, most harmful solutions. But they're also probably the cheapest, which is why they're so scary. All it takes is one or two rogue states seeing this as a viable option, and it might end up happening.