A deep survey of the Large Magellanic Cloud at & SIM;0.1-100 TeV photon energies with the Cherenkov Telescope Array is planned. We assess the detection prospects based on a model for the emission of the galaxy, comprising the four known TeV emitters, mock populations of sources, and interstellar emission on galactic scales. We also assess the detectability of 30 Doradus and SN 1987A, and the constraints that can be derived on the nature of dark matter. The survey will allow for fine spectral studies of N 157B, N 132D, LMC P3, and 30 Doradus C, and half a dozen other sources should be revealed, mainly pulsar-powered objects. The remnant from SN 1987A could be detected if it produces cosmic-ray nuclei with a flat power-law spectrum at high energies, or with a steeper index 2.3-2.4 pending a flux increase by a factor of >3-4 over & SIM;2015-2035. Large-scale interstellar emission remains mostly out of reach of the survey if its >10 GeV spectrum has a soft photon index & SIM;2.7, but degree-scale 0.1-10 TeV pion-decay emission could be detected if the cosmic-ray spectrum hardens above >100 GeV. The 30 Doradus star-forming region is detectable if acceleration efficiency is on the order of 1-10 per cent of the mechanical luminosity and diffusion is suppressed by two orders of magnitude within <100 pc. Finally, the survey could probe the canonical velocity-averaged cross-section for self-annihilation of weakly interacting massive particles for cuspy Navarro-Frenk-White profiles.