I'm linking my savegame to help figure this issue out. Does it matter that it's on a custom map, or is the map info included in the save? The starbase in question is in a Pirate Base grav well.
Anyway, here's the problem. The Argonev starbase seems to fire for a minute or two, and then stop. Sometimes it will start again, but I counted (slowly) at least 60 seconds where the lasers didn't fire at all, despite the fact that there were many ships easily in range. It looks like the other weapons weren't firing either. I tried zooming in, and saw a lot of missiles and such firing AT the base, but nothing coming from it.
Sometimes the enemy fleet would retreat to the edge of the well, only to turn around and attack the base again. The base would always begin firing at them, but would then stop again after a couple minutes. I saw an earlier post that seemed to mention this, but without much detail or a save to try it out.
It says that having ships in-system would allow it to continue attacking. I tried building a Cobalt with the base and then sending it off to the edge of the well to see if just having a ship would make it attack. It seemed to start attacking, and then stop again still. It was hard to identify what could be triggering it to attack again, but I don't think just having a ship there was enough. Maybe having that ship attack triggers it, but I couldn't tell for sure.