there is no general "best method", as it depends on circumstances. maybe you can share more details, like number of stations, are all stations controlled by the controlling faction, black markets available etc. how many bounties are redeemed each day, how much population, how much traffic?
generally:
- reduce the controlling factions influence by missions for all other factions. it is easier to gain influence for a low influence level faction, than for a large influence faction
- while gains are capped, losses are not. do things o actively hurt the controlling faction, like smuggling, killing their ships, shooting their system security
- up your transactions. go bounty hunting at a nav beacon, in uss, or at a res close to station with a kws. redeem your factions bounties only, and everytime you got any claim. it is more about the numbers of redeems, than the value. redeeming 10 times 10K cr is by far better than redeeming 1 time 10 mio CR. similar for trading/smuggling/explo data/ missions
- if the controlling faction has expanded previously, can you bind them in a war or election in that other system?
- can you get a station from one of the other factions in system? if you own a station, you can gain influence more easily by trade or exploration data.
if you haven't read it, read the opening post of this thread
https://forums.frontier.co.uk/showthread.php/193064-A-Guide-to-Minor-Factions-and-the-Background-Sim, and drop your case and questions in there, as you will find neerby all dedicated BGS players in that thread. you don't need to read the few hundred pages before asking anything
in a similar case as yours, a group i'm working with a) brought its own faction timed into boom and "stored" 20 missions for b) put the controlling faction timed into lockdown by security ship shooting c) redeemed the stored missions, and went bounty hunting as above - in that way an election was triggered.