There was some talk a long time ago about going after Extra, but it would mean dropping our influence too much to trigger the civil war and was therefore scrubbed as an idea.
Which is the downside of the BGS, it makes no sense to me that the dominant party in a system would have to drop influence because they want to take over the other bases. It's lime saying Microsoft has to reduce its turnover down to the level of the a small one man band because they want to buy it out lol
One though I had is that when you get enough influence in the system there is a small chance (say 10%) that rather than moving outside the system, that they lay stake to something inside the system. The % needed to take the station could be higher than a normal takeover (because the small faction would be the underdog and engage in things like gorilla warfare or terrorist activities or just be plain tenacious! And it would make it harder for the system to end up being just controlled by one faction).
Also if there is nowhere to expand into the system could start construction of a new station/outpost/port depending on wealth of the faction (again a small chance of it happening). The BGS could also generate a community goal to help fund the construction of the new base. Again the levels could also determine what type of base and what features it would get (refuel, rearm, shipyard, outfitting etc).