The Orion constellation is made up of many stars. Betelgeuse is just one of them. Orion is on the B5 Mongoose Map as being in the Earth Alliance.
Betelgeuse - 427 Light Years from Earth
Bellatrix - 243 Light Years from Earth
So probbly should be fine to use....
Though it should be noted that Orion system isn't in the constellation of Orion necessarily.
Best evidence (the Babylon Project RPG's map, on which the AOG and hence Mongoose's map are based) gives the system name as Ross 6148
However; not turning up much on that name. It is, however on the jump route to Epsilon Eridani (Babylon 5) from Sol, so it will be a close star system to Earth's (within 10 light years or so), and it's almost certainly got the name because in the night sky (of Earth), the star sytem lies within the boundries of the Orion constellation (which Epsilon Eridani does pretty much as well).
Found it - (after a fashion) The Babylon Project's map is a "bit fuzzy" with most of its labels - it's actually Ross 614B
, which is located 13.3 LY from Sol. So a little bit of hyperspace distortion places it a little further from Sol than the Epsilon system, but that's probably the "easiest" way to go in local hyperspace, the direct route from Sol to Eridani (which TBP refers to being investigated at the time) being a bit more turbulent hyperspace wise.
Most of the sytems in the "known Universe" are within 50 LY of Earth (depending upon how much faith you have in the system names given by various sources). Hyperspace travel is fairly rapid though, provided it is along an established beacon route.
Given the distance away, Betelgeuse is probably "off map" past the Yolu, Tal-Konasha and Hyach and a couple of week's (minimum) worth of jumps from Earth.
Virgin territory, go wild with minor (ie New!!!!) League races, but expect a trading presence for the likes of the Brakiri, Pak'ma'ra and Vree.
Quite a good little reference site I found is here...