In a perverse way, this random event has given me an opportunity to do something that the AI could never conceive. I'm going to use the strengthened native influence to help project my influence, instead of just relying on the starbases.
Instead of trying to influence my target planet (Arcturus III) directly while it's in the heart of Arcean space with other planets exerting their magnified native influence, I'm going to bring the cultural border all the way to the planet. To do that, I have to push my culture on the outlying Arcean colonies first. When they decide that I'm more awesome than the Arceans, they will project my influence onto other nearby colonies, and when I get more starbases into position, those colonies will also come to see things my way, and so on. It's like watching dominoes fall, except over the course of weeks and months instead of seconds.
However this cunning plan is all academic right now because even with four influence starbases running Futurama on a watch anytime basis (so, full power) I will barely get a raised eyebrow from the natives. Getting them to revolt is going to require that the Terran Alliance develop new cultural technologies and new franchises, and boldly go where no civ has gone before.
Well actually I see the Korx as being the businessmen of the galaxy, but they're sadly lacking style or substance in this game. Maybe next time, eh Kralax?