I have seen the use of 天の, 天体, 天界 or 天空 but what would be the most appropriate word to use when we wish to use "Celestial" in a way that conveys "outer space" or "astronomical" properties as opposed to "heavenly" or "angelic"? More specifically, which terms are more "physically" related and which ones are more "heavenly"?
天界 is the most "heavenly" word; it's almost always associated with some kind of divine entity, and it's mainly used in fantasy works. 天 is a more traditional word that is usually closer to "heaven", but it can be used as a synonym of 天空 described below.
天空 is not strongly associated with heaven. But it typically refers to somewhere high in the sky where people still can breathe and feel the gravity. Typically 天空の[建物] refers to something like Laputa and Korin Tower. If this shogunate exists in pure outer space (like Death Star), 天空 works but may not be the first choice. You may want to consider other words like 宇宙の, 銀河の, 星間の, etc.
天体 is a scientific term that means "celestial body", which includes stars, planets and comets. It's a well-defined term but it usually does not include human-made structures like ISS.