here is what i'd do.
break down in consecutive steps, so your group and you only have to focus at one goalpost at a time:
Step 1
Push Liberals of LTT 6139, to match your influence, into a conflict with your faction, so you can win Carstensz Laboratory.
Step 2
Win Carstenz Laboratory.
Step 3
Push the controlling faction into a election in one of the two small pop systems they are in.
Step 4
Once the election is pending for the controlling faction, push your faction into Boom using carstenz laboratory. sell exploration data, sell as many different profitable commodities as you can per run.
Step 5
With the controlling faction in election, and yours in boom, you should see gains from random traffic (your faction giving out better missions, and your faction profiting from random superpowerbounties.
if that works, repeat till you match influence and trigger a system control conflict. during that pending, do everything to hurt the controlling faction (smuggling, shootings) and to buff yours.