Having just retreated two more NPC factions (totalling about 8 or 9 retreats done in the last couple of months) the only way that works is to support the lead faction (if they are not the one being retreated) and fail missions for / murder ships from the faction in retreat.
Of course what works for one does not always work for everyone. It might be that the BGS has quirks where you are, or you are opposed directly. As an experiment I induced a faction into retreat by beating it down to 1% and left it while I helped retreat another. Currently after the retreat ended its at 9.3% from passing traffic and BGS Voodoo. From that I concluded that its either an endpoint of missions, missions offered themselves (which themselves are illogical- the mission blurb says we need to retreat, do xyz that adds influence to keep you in the system) or the BGS has a mechanism that adds upward pressure when INF is rock bottom thats causing this.