Not one Object
Having one single satellite to do the job is quite impossible. The smallest single satellite would be way to big. To cover it all, you would at least a surface area of the visible size of earth, whice is around:
Earths Polar Radius^2 *Pi = 126 948 323 km^2
The further you went away from the earth and closer to the sun, the surface your satellite needs to cover goes up, until the visible surface of the sun itself.
Lets assume a thinkness of 1 mm, then this sun blocker had a volume of
126 948 323 km^2 * 1mm = 1.26948323 × 10^11 m^3 = 126.948323 km^3
As you want something sturdy, lets assume they would use aluminum. So the sun blockers weight would be:
126.948323 km^3 * 2,6989 g/cm3 = 3.4×10^11 metric tons
This would be over 5000 years of Earths production of aluminium. Not that it is just heavy to lift up from any gravity well, you need to keep it in place. Even with the Lagrage-Points L_1 and L_2, whice are unstable, you still need correction burns, and that against an object with that high amount of mass.
And, to destroy it simply send something up to ram it. The thinner the shield, the easier it gets destroyed.
Also, sending something so big into space could be detected very soon. Asteriods get tracked with radio and other telescopes. If you had one big satellite, that sends and recieves radio signals and reflects light, it would be easy to detect and conclude, that it must be alien (or otherlings).
So, with similar technologies, it is almost impossible to do.
What about many Objects?
Instead of one collosal satellite, the otherlings could build several smaller satellites with sun covers.
If you want total coverage, you will still need the size and thus amount of material for the sun covers. Benefits would be, you have several small objects, that are easier to handle and to steer, keeping them in a specific orbit would be easier. Also, destroying them would be far more challenging. Even a bigger explosion would only take out some of the satellites, as you could space them for a short time and send them back into the position they should be, then replace the missing ones.
Additionally, you could build them one after another, sending them over a long period of time. Then, if you have them in place, spread the sun covers.
They still would send out radio signals, but much smaller than the collosal one, it would be harder to detect them, but not impossible.
You dont need to cover everything
As stated in your linked question, even reducing the amount of light by 1% would need a really big surface to block.
If you had only a few satellites to block the sun, you could organize them to block the light from some important places. Rural centers like the american coasts, central europe and china would increase the amount of energy needed in those areas. If its a energy starving world as ours, this could lead to devestating conflicts.
Blocking sunlight in areas of food production, plant cultivating and so on would lead to massive starvation.
Also you wouldnt need to cover the oceans or the ice caps at all. Those have the hightes albedo, so blocking sunlight from those spots would only have less effect on the climate. With an armada of small satellites with sun screens, you could organize this very well