I'm a little confused why this is necessary?
I mean, let's stop a moment and thing how freaking AWFUL this would be to stellar dynamics. We're in a society where effective immortality is possible. This is a device that will take centuries to TRULY screw stuff up, but screw stuff up it will. You move a star around, you're effecting gravity wells and stellar dynamics for dozens of light years away, you are changing how ENTIRE STAR SYSTEMS FLY ABOUT and that gravity is going to effect everything it passes by. You are going to throw rogue planets about, you're going to deform how star systems swing around each other in this galaxy, and probably cause them to swing into ONE ANOTHER. Even just moving it someplace else, assuming it doesn't deform gravity wells as it moves, is going to cause this over the course of centuries and people are going to be thinking about that sort of thing because this is something that will effect all of their lives because with ST tech, people are effectively immortal.
It also can be used OFFENSIVELY, what's to stop this from slinging planets, or stars, into enemy star systems?
Why wasn't this used to effectively wipe out the NTX?
Why not use this to move core systems?
Any OOC promise, to not break suspension of disbelief, needs to have IC reasons backing it. It can't just be a promise, there has to be an IC reason for it to make sense for why people DON'T do it above and beyond 'nobody is that big of a jerk' because we have more than enough historical precedent that 'yes, somebody is that big of a jerk'.
The entire thing is a can of worms and a level of power WAY too freaking high for anybody in the setting. This is Kardeschev II-III scale building, many MANY solar systems worth of resources, their entire total energy output, thrown to achieve something like this, for almost NO POINT AT ALL given how fast terraforming can take place, which as I understand it, is a process of a few decades at most.
Why move a star system around? There's hundreds of the bloody things in this part of space alone. If we assume that the map marks approximately a 50 or so light year box (It doesn't, it marks about 200x200ly box) then it'd contain, based on what's around earth, 2000 star systems. With around 130 of those MAJORLY SIGNIFICANT in some form, just from their planets (and that's if it was a 50ly radius -sphere-)
You raise that number to approximately the 4th power, and then some, for the area of space we are currently covering.
There is literally, positively, no possible reason, at all, in any way, shape, or form, this device would be necessary. You'd spend less energy just terraforming planets and moving people to them than you would moving stars around with already inhabitable systems.
Also, and this is important, REMEMBER THE RULE OF TECHNOLOGY! ANY SUFFICIENTLY ADVANCED ENERGY SOURCE IS JUST A GUN WITH A SPECILIZED TRIGGER.
If we've got the energy to move -stars- around, then you have an equal amount of energy to be able to deploy as WEAPON SYSTEMS.
So if I might say. "Screw Yamatai having this. Screw -anyone- having this unless we want to scale this out of military sci-fi and into god-war sci-fi"
edit: To be clear, this is something that isn't even in the area of 'hard sci-fi' versus 'rule of cool', it's a problem of scale and power level.