Do sectors pay for synthetic upkeep

I had a similar problem. I vassalized my rowdy neighbor and integrated him doubling the size of my empire. Only problem was they were a bunch of xenophobic slaving ****s. Try this;

1. Click on the rowdy planet and go to the planet summary screen
2. Check out the edicts list.
3. Happiness and Ethics Divergence.

Happiness will decrease the likelihood that pops join a faction. Ethics Divergence will change the traits of their pops over time. Usually towards your main race's traits.

Give it a few years. Keep pumping up happiness. Change government types if it increases happiness or ethics divergence. If all else fails, station an army there and wait for the revolts. When you crush them, they'll shut up for a while.

Hope this helps!

You need to add a system with the resource to the sector in order for the sector to have access to it. Your core planets have access to any resource in your core sector, which is all the systems not in a sector.

So, in your specific situation, you can either remove the system from the sector so that both it and a system supplying the resource are in your core sector, or you need to add a system supplying the resource to the same sector as the system trying to consume the resource.