any such object placed there would need to have a diameter larger than that of the moon

Well that's kind of my point, that's still a lot smaller than what Elon is suggesting. Elon suggested a sphere with a diameter larger than the earth, if the alternative is a disk larger than the moon, well that actually seems like a much better deal. Also, assuming a disk and a sphere have an equal diameter, the sphere has 4 times the surface area, so that's not a trivial difference.

Lofting something that massive up there and more importantly keeping it there given that it'd also be well within the gravitational influence of the moon would be quite the challenge.

That's interesting. Yeah that could be a challenge. Given the size of the thing, it seems like the obvious thing would be to utilize solar wind for maneuvering, as it's already essentially a solar sail.

The Japanese space agency tested a solar sail in orbit with a novel steering system, rather than changing shape, it used something much like LCD cells to shutter individual quadrants of the sail. Something like that could potentially work.

Point two is that the Deep Space Climate Observatory is currently already parked there.

Yeah, that's a good point. Although if you were actually building something this big out there, you would probably build in some capacity for probes to dock to it. This is a huge installation after all, a facility more than a probe. Or just add on a module that duplicates the capabilities of the deep space climate observatory. I mean once you're constructing something this massive, that additional cost has gotta be a drop in the bucket, right?