I have similar problems in my current colonization. The faction I wanted to move up, had no missions on the bulletin boards because they were so low. Fortunately they had a system a few jumps away where they were at the top, which meant they offered bounties on pirates in the rings, that I could then hand-in in my system. Due to the way the BGS works, handing in small bounties each day, works better than a big bounty on one day, but hey... killing pirates is fun, so my sporadic play times meant I did this (I'm the medium yellow bar that climbs) by jumping back and handing in a few million each time.
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Now we are level it has switched to a war... which is going nicely since no-one else is contending it
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Inara is great for monitoring all this. I think I see your commander, can work out which system you are talking about, and the "raiders" do seem to be in a similar position as my faction was, so this strategy should work for you too. Just remember that winning a war will only gain you a single asset (i.e. station) so you might have to win the war, then work
for the opposition to trigger another war (that you win another asset in) and another war, and another...