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We present design considerations for the Transiting Exosatellites, Moons, and Planets in Orion (TEMPO) Survey with the Nancy Grace Roman Space Telescope. This proposed 30-day survey is designed to detect a population of transiting extrasolar satellites, moons, and planets in the Orion Nebula Cluster (ONC). The young (1-3 Myr), densely-populated ONC harbors about a thousand bright brown dwarfs (BDs) and free-floating planetary-mass objects (FFPs). TEMPO offers sufficient photometric precision to monitor FFPs with ${\rm M}\geq1{\rm M}_{\rm J}$ for transiting satellites. The survey is also capable of detecting FFPs down to sub-Saturn masses via direct imaging, although follow-up confirmation will be challenging. TEMPO yield estimates include 14 (3-22) exomoons/satellites transiting FFPs and 54 (8-100) satellites transiting BDs. Of this population, approximately $50\%$ of companions would be "super-Titans" (Titan to Earth mass). Yield estimates also include approximately $150$ exoplanets transiting young Orion stars, of which $>50\%$ will orbit mid-to-late M dwarfs and approximately ten will be proto-habitable zone, terrestrial ($0.1{\rm M}_{\oplus} - 5{\rm M}_{\oplus}$) exoplanets. TEMPO would provide the first census demographics of small exosatellites orbiting FFPs and BDs, while simultaneously offering insights into exoplanet evolution at the earliest stages. This detected exosatellite population is likely to be markedly different from the current census of exoplanets with similar masses (e.g., Earth-mass exosatellites that still possess H/He envelopes). Although our yield estimates are highly uncertain, as there are no known exoplanets or exomoons analogous to these satellites, the TEMPO survey would test the prevailing theories of exosatellite formation and evolution, which limit the certainty surrounding detection yields.