Let's assume that the UFP are able to keep the star under control and the Dyson Sphere in check so that it stays at a stable orbit/rotation/equilibrium position around the Star.
Then what do you do?
The UFP DID demonstrate the ability to be able to reignite a star (happened on DS9). Apart from that, there was also an attempt to stabilize a dying star in TNG with a torpedo... which unfortunately didn't work at the time (it seems to me the UFP may have learned from that practical attempt until it culminated in the ability for them to re-ignite a star - which seems more difficult compared to an ability to stabilize one).
All in all, considering UFP encounters with stellar phenomena leading up to the late 24th century, I wouldn't put it past them to be good at it.
Also, the star in the Dyson Sphere seemed more or less stable apart from occasional 'hiccups'... so its possible that the problem is not 'extreme enough' to be beyond UFP ability to repair/stabilize.
I think the first order of business would need to be getting a handle of the Sphere's operations... means SF would need to find a command center (if the sphere has one), or create an interface with which to interact with the Sphere. Since there's a door there and tractor beams, presumably, there is also a computer interface (or the computers could be scattered all throughout the sphere's infrastructure networked together (this is kinda what I was hinting at when describing the construction of UFP's Dyson Swarm - aka, since each solar collector is 10km big, the inner side of it would contain say 10-20 Intrepid class computer cores, same amount of transporters, industrial grade replicators, tractor beams, antigravity controls, thrusters, field/shield generators, and impulse engines [starship type], pattern buffers, sensors/communication systems, subspace technology (you can also throw in fusion reactors across the collector as a backup - but no real need for M/AM generators since solar would be more than enough)... all of those would be interconnected/networked to share resources and effectively enhance overall replication/materialization capabilities (so in a sense you get the technology of a starship, just scaled multiple times over and networked across 10km inner surface area which wouldn't need be more than 1 deck high either).
The outer section of the collector (which faces the sun) can easily house habitats etc... all made from the composite material which accumulates solar energy, so it would never run out - also, we already designed solar panels capable of working at night (harvesting the infrared), so the same would (probably) be accurate for UFP solar technology (which would be hundreds of years more hyper advanced to the point where even a small surface area would be able to power a building - now imagine lining your hull in the material in question - woof, the harvested energy would likely range in many times over compared to what a single solar collector/habitat would ever need).
Also, a Swarm or a Sphere DOESN'T have to be made of Neutronium... and UFP has several materials of choice (even dating back to the 22nd century) which they can blend into a new composite and used as a baseline for the swarm or an actual sphere if Neutronium is out of the equation in terms of production (for the time being).
So, UFP DOES in fact have the ability to make their own Swarm or a Sphere... either one would do the trick... its just a Swarm would be far easier to construct (and less susceptible to impacts from asteroids, etc... but that's easily remedied with a simple integration of a deflector array). For us (in real life) it would take us maybe 20 years to make the swarm if we used automated self-replicated bots which house 3d printing, molecular manufacturing (since 2015) and atomic scale manufacturing (since 2018) technologies (and simple extrusion technology existed for decades - it predates 3d printing).
For UFP in the 24th century, replicators convert energy into matter... that way, they SKIP the process of accumulating raw material and just replicate the Swarm or a sphere in sections per each materialization cycle depending on how long it takes to recharge the capacitors (since its being gotten from the star) on the spot - this would be a huge time saver (along with the fact you just materialize matter - as opposed to fabricate it on the spot).
The existing Dyson Sphere (from TNG) however might be defunct in some parts... and because its made of Neutronium, it could demonstrate difficult to interact with and repurpose as a result (at least for the time being).
But, to me this never made much sense since the UFP DID encounter a Planet Killer which they disabled about 70 years earlier and was made of solid Neutronium. I would postulate the UFP would be able to adjust their technology to be able to penetrate and interact with the hull which would give them more info on how it was made, and the other 50 years would be more than enough time to start making their own Neutronium... and because UFP is comprised of dozens of species in the 23rd century (and technology and science evolve exponentially), the UFP would have been able to crack this in far less time... but since the writers decided to ignore this completely... its a bit moot.
In my head canon (or at least, alternate reality, UFP was much more advanced than the Prime Timeline thanks to all of that).
UFP would likely use the Dyson Sphere for study... possible colonization, and it could be used for scientific experimentation (given how much energy that thing has at its disposal), and starship construction.
They could literally replicate whole fleets of ships into existence... hundreds of thousands to millions... if not billions.
Obviously, replicating that much ships in a single go (while possible), wouldn't be of much use if the UFP cannot crew them... but automated ships would present a different option in that case.
So, my guess is that they would use the Sphere to supplement starship construction over time (replicating how many ships is needed), but they could still replicate a ridiculous number of FTL multispatial probes (like the one VOY designed) to send into far reaches of the Galaxy ahead of starships.
The Sphere (or a UFP made Swarm/SPhere)could also be used to integrate Hypersubspace and Astrometric sensors together... they could massively amplify range, communications, real time sensor data (I mean, if regular Astrometric sensors already can scan 2500 Ly's away on an Intrepid class ship, just picture how far out a system the size of a Dyson Swarm or a Sphere could scan in real time... probably throughout the entire galaxy - or most of it... it could also be used to maintain real-time comms with species throughout the entire galaxy in real time - and it could probably be combined with subspace transporters or Tranwarp beaming (which is ever elusive in the Prime Timeline even though it was supposed to have occured by now) for transfer of objects to distant locations in the Milky Way (which could be one way of maintaining contact with species across the Galaxy - and heck, even humanoids can survive supspace beaming process if its treated on the spot - at least until UFP finds a way to protect live tissue - beaming of automation, tools, whatever... which could effectively start terraforming of other planets, or even remote construction of starbases and other Dyson Swarms or Spheres - just beam self-replicating technology near a star and instruct it to start building).
The sphere could be used to further research into Soliton Wave technology, Quantum Slipstram (and the power source it used since it was non antimater based) and Tetryon Reactors. Even if at this juncture the Soliton Wave is still not viable to be used for Warp travel without warp engines... aspects of the technology could be used to increase efficiency of existing Warp drives by quite a lot (it was mentioned during the initial test that there was less than 2% power loss between the ship and wave - increasing efficiency by at least 400%).
If this could be adapted on standard warp technology, then that means, SF would be able to access higher Warp velocities with minimal power drain.
It's also possible that the USS Prometheus (the 4 nacelle ship with multivector assault mode) already used a version of this technique (along with far better SIF technology) seeing how it was able to maintain a speed of Warp 9.9 without any bumps or problems (unlike VOY which was only able to maintain 9.75 for 12 hours) - which means it was travelling at 21 473 times speed of light towards Romulan Space. That actually opens up massive exploration capabilities of the Milky Way using regular Warp speeds already after VOY's return, and even dwarf galaxies in the immediate vicinity of the Milky Way are accessible to UFP with just Warp 9.9 (but I think Warp 9.91 would have been possible to sustain for long periods of time, which would be double the speed of 9.9).
At any rate, UFP making its own Dyson Swarm or a Sphere, or repurposing the existing Dyson Sphere can be used for numerous things... from scientific/technological R&D to exploration, making its own Quantum Slipstream drives, developing new power sources... real time scanning of the Milky Way and establishing intra-galactic communication (possibly even extra galactic)... anything really.
The possibilities are quite large when you have at least 384.6 Yottawatts to work with (and that's just our own star). This easily trumps M/AM and dlithium in terms of energy output of a Warp core by orders of magnitude... or tens of thousands of Warp cores (I think).
Making your own antimatter, mining dilithium for regulation of reactions, etc... is a bit pointless on a stationary superstructure like the Dyson Swarm or Sphere.
Why bother when the energy being emitted is already there? You just need to harness it... and making enough photovoltaic surfaces for the Dyson Swarm habitats or an actual sphere (when powered by the star itself) is probably a lot easier than to make independent M/AM cores and mine dilithium - which can still be done for ship... but otherwise, not needed for a Swarm or a Sphere.