The size of the save file is obviously dependent on a number of factors: scripts, units on map, buildings on map, etc. etc. We expect the file size to increase as the campaign progresses simply because the player and AI factions have more stuff around the map. If the game is saving character data for all retired agents, then we expect each save file to be larger, i.e. we still expect a linear dependence but with a greater derivative. In principle, all we would need to do is play two campaigns up to some turn number exactly the same way (hands off), one with the named forts enabled and the other with the script disabled. But since I'm going to have to manually click through all the turns anyway (the script won't fire if I give control of all factions to AI in hotseat and I'm afraid that might affect the save sizes) I may as well just make a graph.
I've only implemented 8 named forts so far, so I collected a reasonably large data set to make it easier to discern any slight different in slope. If the game is indeed storing data on all the retired agents, then we expect the trend line with the script enabled to have a greater slope. As shown below, there is no discernible difference in the save file size trend with the script enabled vs. completely removed from campaign_script.
If anything, the data shows a slightly greater slope for the campaign when the script was removed but I chalk this up to statistical anomaly. The game adds an average ~20 KB to the save files each turn, at least for the mod I used to test.