would be to kill the non-anarchist factions ships, do I have that right?
basically yes.
now, there are several things to consider:
a) fehu is a system with (probably) a lot of player traffic. e.g. all factions are permanently "backed" by all players
b) influence and state both have an effect on mission generation
going from there, you could either
- put the anarchy faction into a state, to generate different missions. problem with that is, they will be permanent backed by other players running missions. but killing that anarchy faction ships till they get into civil unrest or lockdown by shooting their ships (going bounty hunting in system) could work. once they are in lockdown/civil unrest, they will offer other missions. problem is, you will probably be hostile with them then. on the other hand, you will be in system a long time, and can dock regularly for rearming - e.g. you can check the mission board often.
- get the anarchy faction into civil war with another faction, by influencing the other factions influence. if all factions are backed by other players (without intention - just by running missions), it should be possible to make a difference by focussing on one faction only. pick one with less influence, and back them - or pick one faction with more influence then the anarchy faction, and kill this faction ships - a lot of them. unfortunatly you will get wanted, which will make things complicated. but you want to work for a criminal faction, right? obviously it would be better, if you would do that coordinated with a group of people.
- as soon as this other factions influence matches the influence of the anarchy faction, a civil war will trigger, and you a) have the anarchy faction in another state, that will spawn other mission. yay! b) you can gain reputation by cashing in combat bonds.
without a faction having a station (with a market, a blackmarket, and/or universal cartographics) there isn't a way to influence that faction outside of combat activities or missions.
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looking at fehus stats, i guess the "most easy" way would be, to raise Independent Fehu Liberals influence, till it matches Fehu Silver Rings Influence. Independent Fehu Liberals are currently in lockdown, and you could help them to get out of it by bountyhunting in Fehu with a KWS, and cashing in bounties only issued by them and superpower bounties at their station. They own a Surface Port with Market, so you could try to back them after lockdown has ended by a) running missions b) handing in bounties issued by them - and only those c) handing in superpower bounties at their station d) sell goods at their station. in Fehus case: go mining. e) sell exploration data at their station.
once they match influence with Fehu Silver ing, you will change sides and step your liberal allies into the back ;-)