That still seems to be reaching a bit and here's my reasoning:
1) The sleeper ships couldn't have gone far. Their level of technology for intersteller travel is not that extensive-- 800 years later they only managed to explore part of a local cluster of stars. If their transport technology is more advanced, we would have seen the game arena span a major part of a galaxy or galaxies.
2) If they haven't gone far, then they are still within the region of the Milky Way near Sol. If that's the case, why wouldn't the colonists be able to decide on a name for the region? The Greeks of Antiquity already did it for them-- They already named all the star clusters and regions visible from Earth out to a couple hundred light years with names like Orion, Cassiopaea, Taurus, Gemini, Ursa Major, Sagittarius, The Pleides, Sirius, Polaris, Pollux, etc.
3) Why use "Sirius" as a code-name? There are better, non-descript code-names that wouldn't telegraph their intentions like the name of another star system would. Something like say "Shangri-La."
4) I'm sure the Coalition was tracking them all the way out of the solar system and noted their bearing and speed. That would have made the use of a distractive code name moot I think. "Comrade General! The Alliance Fleet is heading for the general direction of the star system Sirius. Shall we pursue?" "Nyet. We will consolidate our control over this solar system first before we decide whether or not to hunt them down."