Other posters have focused on two large asteroid belt clearing problems: excessive time and excessive energy are needed to suspend disbelief for the genre of hard science fiction.
Excessive need for space or time seems the lesser issue in a Einsteinian universe.
To suspend excessive space-time, I agree with brichins' idea of using wormgates, a SF technology that harnesses macro wormholes (as opposed to quantum wormholes that seem closer to present physics theory).
I agree, in validly suspended SF, that each end of a sizable wormhole can be placed where needed to move even large planetoids like Ceres. I think all asteroids could be teleported adjacent to each other, with matching momentum for bolting (stone) and welding (iron) into something(s).
Something(s) 4% the size of the moon per Arkenstein XII.
I see no obvious SF reason why wormgates can't teleport other wormgates in a preprogramed pattern around the asteroid belt. What I might call leveraged teleporting, starting with a fleet of cargo starcraft powered by (consensus SF) hyper/warp drive.
Teleport clearing the whole asteroid belt certainly wouldn't be instant, and could take years, because electrons and photons - even in the peripherals of quantum computer circuits needed to solve such an enormous set of calculations - travel at about 95% of light speed (in copper).
One will need a great many individual computers/processors/hardware, and quite at lot of (consensus SF) subspace signaling to project managers in, say, a triangle of bases orbiting the solar system. Debugging committees, even with the help of AI, will have to solve many initial project design issues. That's not going to change in the future.
Mark's delightful analysis of the excessive energetics seems more troublesome.
Einstein seems to allow the cheating of our ordinary calculations of space and time, but not the cheating of mass-energy conservation. Mark probably can calculate how much energy it takes to move objects into higher orbit, if it's 4% of the moon mass, raised from the asteroid belt, to spiral orbits into Jupiter for slow disposal. Jupiter disposal has its own set of problems, but I've read that Jupiter is too small to be set on fusion fire.
I don't know how much recoverable, reusable energy, and lost, heat-dissipated energy it takes to operate a wormgate.
Even if you recover most of the reusable energy after the worm field collapses, to inflate the field, one still has to store a great deal of energy joules in pre-charged capacitors (sized 10^xx farads?).
For story purposes, I think the average reader better comprehends kilowatt-hours (at 20c each) to the equivalent physics joules. Will the Emperor pay for the heat-dissipated energy?
The good news is that your story is premised as tongue-in-cheek SF like Hitchhiker's Guide to the Galaxy. That genre depends more on clever writing than a serious SF suspension.
You've mentioned a bureaucracy, so your story could be one of massively bungling the project, yet in Homer Simpson fashion, all the bungling cancels itself out to result in an unexpectedly successful conclusion.
One way to compress 10 years into a short story is to tell it as a series of dated historic documents, like royal proclamations, emails, radiophone transcripts, corporate invoices, lawsuits, arrest warrants, and news accounts.
You could cover over story holes with puzzled notes from a future historian, writing that he can't find documents to explain how event X occurred improbably - which works with tongue-in-cheek SF.