So I'm up to turn 1500 now. I've made the following changes:
* I invaded 3 of the 4 planets so I can focus on the last one.
* I control a nearby planet, which I have loaded up with Influence stuff (7 outreach centers, 4 capitals including a very well-placed Entertainment Capital for the adjacency bonuses)
* The area now has 7 fully upgraded cultural starbases, 6 have my planet within its border, 5 have the enemy planet within its border
* The planet is generating 268.5 influence per turn (base influence goes up 26.85 per turn)
* Current Base influence: 17778.6
* Total with bonuses: 43557.6
Theoretical upper bound for enemy planet:
* 6.85 per turn for 1500 turns = 1027.5 base influence
* Assuming 1 + 200% bonuses (very generous as mine fully loaded is only 1 + 195%) = 3083.5 total influence
So my planet's influence is over 14x the assumed theoretical max of the enemy's. There may be some decay so the local influence is probably less than that, but not much. Something is wrong.