I am peaceful and don't extort my neighbors, so I can't help there. Why not try? :) You can even cheat and save/reload.
More fuel cells, and improved storage tech (fuel cells) are of course tools for better range. As are more efficient hyperdrives and more efficient reactors. Many things are involved. I usually run with AI designs, it is one way of not making things too easy.
Shooting uses energy. To get energy the reactor burns fuel. Using weapons with less energy range helps real range when it is a question of "travel - fight - refuel". And long travel is usually due to fighting, right?
There are certain actions that are very efficient and of no help in keeping the game challenging. You may notice if you do any of them yourself. Reading the quick strategy guide on this forum should give a nice overview...
Resupply ships are lovely. They are powerful warships, the AI equip them with long range scanners, and they function as efficient gas mining bases with lots of docking ports (for faster refueling of fleets). They are not trading points, and your state ships are the only that will ever refuel there. They also lift only caslon and hydrogen even if other resources are present. One issue with gas mining stations can be that when there is fuel problems, all fuel mined is quickly reserved by freighters that want to bring them to a spaceport. These reservations can also be alien. This usually makes gas mines unable to refuel a big fleet, since there is not enough "free fuel".
My tips are: To use them, they need to deploy. Locate a fuel resource (cloud/planet), have the resupply ship selected, and right click (or ctrl-right click) the target and select "deploy". After some time, the ship will be deployed, and start mining gas. When deployed your ships can refuel at it (like they do at a mine/spaceport).
What I do is that when at peace, I dock them at a good source (high percentage) that only carries the fuel I use. Caslon or hydrogen. Or that has a high percentage for what I want, or low for the other. This fills its stores, meaning ships can refuel once it docks, and don't have to wait for it mining new gas. It does not hurt if the chosen spot is one where you could possible need some extra fuel either...
When you plan war, use the galaxy map "known resources" filter to locate your destination. You want a reasonably close (and good) location with your fuel. I tend to try avoid "busy" systems, if I can choose. Sometime I attack enemy systems and park the resupply ships inside. I have deployed on the gas giant the enemy capital orbited, to refuel while fighting its defenses...
They are also a blessing if you have large fleets, as those will run even the best spaceport dry if they have to refuel often and arrive with little fuel. In peace I may park them at my edges where fuel supply is scarce, to provide fuel for the local pirate buster fleet.
This a screenshot from my Yor AAR, where I (purple) struggle in a war with the reds below me. Massive battles went on in my southern parts.
At the bottom you see my resupply ship (long range scanner) setting up in a gas cloud. A bit above is a fleet that has busted an enemy system. It is at the limits of what I could attack from my southern colonies. With the resupply ship as a base, this fleet can harass enemy systems both to its north, east and west. If the fleet runs out of targets, the resupply ship can move.
There was a crazy fight in those southern areas of mine. The image is after grabbing one enemy colony and getting some sort of control. 4-5 large fleets worked full time to protect my colonies, and even this was core areas for my empire, fuel supply became shaky. In those situations a resupply ship or two in the south (inside my borders) could be useful to help the fuel situation. They could even park on fuel sources in the systems being defended.
Oh, and when you struggle to deploy in a gas cloud, make sure you click the cloud itself and don't try to deploy on the system-level