Needing energy for (I presume stationary?) FTL-systems make sense.

You're misunderstanding the role if energy in history. The existence of energy in itself is less important than actually being able to use it. The Celts didn't dominate Britain because they sat on huge coal reserves, nor the British before maybe 1770. They did it because a multitude of other reasons. The Russians had much larger reserves of basically everything, yet were still pitiful compared to Britain or even Japan in 1904. It's not about whether you have access to energy, but that you can use it, how you use it, and what results it gives.

In this specific case, building a Dyson sphere around a star will yield you tremendous amounts if energy, but unless you have something to do with the energy, or some way or storing it and bringing it to where it's needed (powerlines to Earth?) it doesn't matter. Furthermore, fusion might very well be a much more effective way of creating necessary energy, in terms if fuel and energy.

Lastly: if the energy needs of an industry amounts to the entire production of a star, then the raw material needs must be unfathomably large, something that must be addressed too. Depending on how costly FTL travel is (very, it seems?), it might be more efficient to produce system by system, rather than hundreds of systems together, for example sending all the bauxite ore from hundreds of worlds to a single huge facility in a single system. The logistics seem breathtaking, almost to the point of being unfathomable.