As Starwolfe indicated, only the combat bonds have direct influence, the missions, rather all/any missions have no effect other than to draw in outsiders to fight and hand in bonds.
Also very correct in the murder aspect, as in away from the CZ's killing clean ships will push killed ships faction towards civil unrest as well as deflate their inf%, by combining this tactic along with normal tactics you can actually push past the normal max 10% daily influence change and easily hit a 20% influence change if done right.
On top of that, other factions tools still work as long as they are not in war or civil war, so by adjusting their inf%, the warring factions INF will also change.
Keeping in mind in all this that anarchy factions if present(usually are) don't have clean ships...and even though killing them tarnishes your rep with them, it actually raises their inf...
So the main answer, if you are trying to control the INF of one of the warring factions, don't waste your time taking missions to do it, just get in there and kill the oppositions ships, hand in bonds often for best impact.
easier to be more helpful if I knew the pop size, and also wether you simply want a quick win, or if you want a prolonged war, as a win or quick win is not always what a person wants.
But if after the first day or 2 of the fight, the odds are not yet in your favor due to various reasons, you may then want to also resort to the killing of clean ships outside of the CZ if its possible, it will make you wanted but it will give faster results. If you do this though, thanks to the new C&P, be aware of the superpower ATR ships, once they start showing up....they can take down your shields in an instant and waste your ship in another instant, it is not a fight you can win, you either run or die.
Knowing the value that a bond amount has against a particular population size is vital for precision, but precision is not always required.
if the pop is in the billions, it will be a large amount and will be hard to affect change, especially if there are others taking part that you may or may not be aware of.
If the pop is below 1mil it will be easy to affect change.