Assuming that the screen is electromagnetically "perfect", in that it reproduces exactly the radiation we'd expect to observe, including parallax AND spectra, with the required emission and absorption lines, all appropriately redshifted (quite a difficult challenge if you ask me), the answer is probably...
Although as @benrg correctly pointed out, we had been detecting cosmic rays a hundred years before this, and neutrinos from the SN 1987A supernova in drum roll 1987, we'd be definitely suspecting something by this point.*
This is the date Voyager 1 crossed the heliopause and took direct measurements of the interstellar medium. This is the area dominated by particles coming from the other parts of our galaxy, which would behave noticeably differently, were the Sun to be found not to be moving at breakneck speed through that medium, creating a bow shock and everything. In fact Voyager 1 data would've indicated that something was amiss years earlier, when passing the termination shock layer.
But even if we assume they also installed massive interstellar fans to simulate the particle flow, we've got...
This is the date of the first non-electromagnetic measurement we've made of our universe, when we successfully detected the gravitational waves from a black hole merger.
Now the game is up, that screen may simulate any electromagnetic or particle radiation, gravitational waves are a whole different ball game. There is no known way to fake them with a projection device.
Of course if they went to the length of installing a screen so vast it'd weigh more than most stars, why wouldn't they do this too...
* There is an interesting Earth-based corollary to this. We have some fairly credible hypotheses about how cosmic rays are seeding lightning in storm clouds. If they are correct, we'd be seeing a difference in how thunderstorms behave on our planet. Of course we wouldn't have anything to compare it to, so we probably wouldn't discover this clue until after some other clues were put together but I thought it was worth mentioning.