A more detailed but still not exact explanation: My thought was to have a monitor on faction turn start, reducing influence. Sample figures are fuzzy due to the authority limiting the control, but they could beI could for example code something like only half the Influence gain for settlements you controlled from the start (removing the bonus altogether would underhand large factions and be over-compensation in my opinion). Actually I think that might be what you meant in the first place.
*if the empire have 5 or less settlements, it decays by 0.01 (leaving 0.05-0.29 of the settlement bonus)
*if the empire have 5 < settlements < 15, it decays by 0.1 (leaving roughly 0.2-0.6 of the settlement bonus)
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*if the empire have more than 50 settlements, it decays by 1 (leaving roughly 0.2-0.4 of the settlement bonus, plus 0.02 per settlement above 50)
So it complements the settlement bonus, really, and leaves the profound effect the authority has through firmer control over more settlements. Without any form of check, the big empire would gain as much just sitting still as a smaller one which gains a settlement and wins a major battle too. I would think the latter would get the more publicity in that time. But your other suggestions would make sense as well. Just a suggestion, and would of course require tweaking of the values
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