The GTA was headquartered in Sol because that is where humans are from. The VPN was stationed in Vasuda Prime because that is where the vasudans are from. Both governments were "moved" during the great war (VPN to Aldebaran, GTA to Delta Serpentis). The GTVA is the combination of both of those governments, so it makes sense that it would be in a neutral place (Beta Aquilae). Just because Sol is reestablished doesn't mean that that is where the GTVA would move to (unless the Vasudans became greatly diminished and needed the protection of humans).
Also, even given a stable Sol, their technology would be behind the GTVA. Sure, the Lucifer wreckage would help, but they still would be behind. I think that there would be a brief skirmish at the most, maybe driven by the leaders who didn't want to give up power (kinda like a Denethor/Aragorn situation in LOTR). At the end, there would be the GTVA that is bolstered by the reintegration of Sol.
P.S. On a side note, wouldn't it be cool if the Sol technology somehow was superior to the GTVA and they reopened the jump node to delta serpentis first?