Some basic problems with the concept I see are :
the Earth Federation wants to deprive the rebels hiding among the moons of the gas giants of solar power
It seems very unlikely that 1000 years from now they'll be relying on solar power (which is very weak anyway at that range). We expect to develop fusion power within the next 75 years or so and the "rebels" will be orbiting gas giants so no shortage of fuel.
Dyson Sphere to block out the sun for the gas giants
Ignoring the fact the won't need solar power in the first place ...
This does not require a Dyson Sphere. A Dyson sphere proper would require an insane amount of resources and it would be easier to build billions of robot warships to hunt the rebels down and wipe the rebel scum from existence.
Instead you "just" need build a few large shields that can shade the gas giants and their moons. You place them in powered "pseudo-orbits" (real orbits would have the wrong orbital period, you need to overcome this with active orbital maneuvering). These would require a lot of resources, but still a lot, lot, lot less than a Dyson Sphere.
At most you'd build a ring, although like a Dyson Shell these are unstable in orbital terms and not practical.
Dyson Sphere top completely block out the Sun.
That's a Dyson Shell. These are not stable in terms of orbits. The mechanical stresses are also enormous (even a ring has enormous forces on it, but a sphere is just a disaster). Modern interpretations of this tend to be Dyson Swarms and their variants.
The high ground
You don't need mines, you need something to protect you from the vast numbers of extremely high velocity, low mass kinetic weapons the rebels will be flinging at you from the high ground they occupy.
Gravity works in their favor and they will have fusion powered engines, so they can almost certainly fling relatively small mass items at you with extreme accuracy and enough energy so that any one of them would be enough to wipe out all life on e.g. Earth. They can just wander off to the Oort cloud or Kuiper belt and they've an endless supply of junk to send your way.
Again this makes building the billion ship robot fleets and hunting the rebel scum down that much more important !
Do it to them before they do it to you !
Waiting around trying to patiently build a defensive wall (your Dyson Sphere) is pointless as all it does is given the rebel scum (did I mention they're scum :-) ) time to attack you with kinetic weapons. Better to go on the offensive - and more fun anyway.
So you build vast robot fleets with the same tech you were going to use to build a Dyson Sphere. These patrol aggressively and in particular you saturate the gas giant systems with even higher numbers of these.
You build kinetic weapons that can do them damage, perhaps even wipe them out (lousy rebel scum) before they do damage to you.
You do not need to wipe them out, of course. You can just suppress them to the point that your robots fleets can keep them under control and slowly reduce them to nothing.
Now back to ...
Yep, after all that, there is a use for mines.
The rebels are going to use mines (of a sort) to defend against your vast robot fleets. These mines will need to be passive arrays triggered by the detection of nearby fleets. They're going to have to use weapons fire - so not a minefield in the sense you may have meant. They will have limited numbers of projectiles, but these can be autonomous missiles with high speed and targeting capabilities. They can employ very high yield thermonuclear warheads - there is no theoretical limit to this beyond the limited size of the warheads.
Mines won't eliminate a fleet, but they should reduce the numbers enough to allow your own smaller fleets to engage them successfully.
This would be a war of attrition between two forces. One has the mass of the inner solar system, but the rebels have the mass of the gas giants to work with. If the rebels can survive long enough to develop the required self-replicating robot factories and fleets, the inner solar system looses, as it cannot complete with the production potential it faces.
That's why the inner systems need to build those vast robots fleets ASAP and launch a first strike on the rebels to prevent them developing that capability.
But this is also why the rebel scum need to build vast semi-intelligent minefields. Not to guaranteed protection, but to remove the worse of the danger and let them develop their industrial base to the point it can win.