Given how deep it sounds like the lore rabbit hole goes with this, are there any mentions of places like Vega 4/New Shanghai, San Drad, Ryushi, Bunda or any other locations related to the AVP side of the Alien EU?
No. The license was Alien/ Prometheus only and the bible that was used as one of the sources didn't contain any AvP locations.
The RPG references Odobenus Station, which is from AvPClassic.
Necro-quoting. That was a mistake. There's only a single reference to Odobenus. The other mention of a station in orbit is given a different name. Not sure if that was fixed in later printings?
Just took a look at my pdf copy of the core rules and no, that's still how it's listed; as an orbital Xenomorph research station around LV-426 in two instances, but later with that same station being named as Kathar Station a "a minor research platform maintained by a custodial staff" (it's also listed as Kathar Station on the map, with no references to Odobenus there).
That makes me wonder about Chigusa Corp being included in the list of corporations in the core rules then too, unless it also cropped up somewhere in the Alien EU without the AVP connection?