Just to update:
Using some of the advice in this thread, I can confirm that killing system authority ships has been extremely effective.
The controlling faction has seen huge drops each day, despite being a high population system with large amounts of passive player traffic working in their favor.
Basically, I have accomplished in a few days what weeks of influence-mission and trade grinding could not, and with far less effort.
As to the massive bounty: I’m simply ignoring it for now. Once the system flips the bounty will be moot, although I may eventually use @Rubbernuke ’s strategy of selling off the hot ship just for safety.
Its a lot of fun using a disposable ship, as its a reversal of engineering in a way- G1 is easy and cheap (but the max you really want to go), and tech broker Shock Cannons actually have uses. It also hones your danger instinct and makes you assess targets much more.