Not really, it's only about 40 meters in diameter. With trillions of dollars at stake, I imagine they'd find a way to bury one under lower Manhattan.
I've found some figures here -- this is an experiment which creates a focused neutrino beam with an accelerator, and sends it through 810 kilometers of earth to a distant target:
If I'm reading Table IV correctly (page 26, and I doubt it), their expected signal rates are up to 10^3 counts per [(megawatt beam power) * (10^7 seconds time) * (kiloton detector mass), for μ-neutrinos. Some reasonable parameters (skimming in the article) are on the order of 1 MW beam power and 10^2 kilotons detector mass, for a theoretical maximum of 10^-2 counts per second. (But I'm not sure if the accelerator can run continuously, or just in pulses). For a 10,000 km beam the signal rate would up to 100 times lower, because of quadratic beam divergence (though attenuation is negligible). So that's 10^-4 counts per second. To send 10 bits (as on/off pulses) in 10 ms, you'd need a lower bound of 10^3 counts/second. That's 10^7 times more than this experiment. So basically feasible, if you have the resources of a hedge fund: scale the total beam power to ~3 gigawatts (by linear extrapolation ~$300B, but probably much less), and the detector mass to ~300 megatons (of liquid argon?) (also ~$300B by extrapolation. This about 6,000 times Super-K, or a cryogenic sphere ~3 km wide. Or an array of smaller spheres).
Besides, I'm certain my numbers are large overestimates. I extrapolated numbers from a completely different scale; surely optimizing for this problem would yield very different designs. Like more focused neutrino beams. They have a large fraction of their neutrino beam going out >6 km off-axis (at 810 km distance); at 8,100 km, this would be >60 km off axis. So there's maybe 6-7 orders of magnitude potential in designing a lower-divergence beam.
I've found some figures here -- this is an experiment which creates a focused neutrino beam with an accelerator, and sends it through 810 kilometers of earth to a distant target:
http://nwg.phy.bnl.gov/~diwan/nwg/fnal-bnl/report.pdf
If I'm reading Table IV correctly (page 26, and I doubt it), their expected signal rates are up to 10^3 counts per [(megawatt beam power) * (10^7 seconds time) * (kiloton detector mass), for μ-neutrinos. Some reasonable parameters (skimming in the article) are on the order of 1 MW beam power and 10^2 kilotons detector mass, for a theoretical maximum of 10^-2 counts per second. (But I'm not sure if the accelerator can run continuously, or just in pulses). For a 10,000 km beam the signal rate would up to 100 times lower, because of quadratic beam divergence (though attenuation is negligible). So that's 10^-4 counts per second. To send 10 bits (as on/off pulses) in 10 ms, you'd need a lower bound of 10^3 counts/second. That's 10^7 times more than this experiment. So basically feasible, if you have the resources of a hedge fund: scale the total beam power to ~3 gigawatts (by linear extrapolation ~$300B, but probably much less), and the detector mass to ~300 megatons (of liquid argon?) (also ~$300B by extrapolation. This about 6,000 times Super-K, or a cryogenic sphere ~3 km wide. Or an array of smaller spheres).