Problem 1, do the Klingons only have one world? I thought this was an Empire.
Rome was only one city. The heart is only one organ. If the plurality of the Klingon population is located on Qo'noS, losing the planet is pretty bad. Never mind that, as the capital, it'll have the most important Klingons, and vital infrastructure, and the sentimental value, especially to a civilization that values, if not obsesses over, history as much as the Klingons do.
So, yeah, pack up and move. It's a planet, not a cruise ship. Moving billions of people, and rebuilding infrastructure, trade and supply lines, industrial capacity, and so on, is not an inconsiderable feat. We'll come back to this in a moment.
Problem 2, their atmosphere? what about tidal disruptions and earthquakes?
What about them? The immediate devastation wrought by Praxis exploding was no doubt severe, but it was also instantaneous. Rebuilding after that and recalculating the tide-tables for the Klingon fishermen (it's the hunting of the seas!) is straightforward. Klingons are probably great at rebuilding stuff, considering how much their society values combat and war.
The fact that the atmosphere is in a cascading failure that will render it unbreathable within a lifetime is the more critical problem going forward. Not going to fix that with some old-fashioned Klingon stonemasonry.
Problem 3, what does the size of their military budget have to do with anything? Or are these the same people that wrote this that think militaries are only good for shooting things and not transporting people in all of those ships, each with food stores and medical equipment, no matter how rudimentary it is on Klingon vessels.
It's a drain on resources.
Guns or butter. Every Bird of Prey launched, every disruptor pistol built, every bat'leth forged signifies a theft from those who... breathe and... don't have oxygen due to massive ecological collapse caused by total ozone layer depletion. Ike said it better.
Those Klingon ships aren't just floating around for show, so they can do an exciting flyover for the Day of Honor parade, waiting to be converted into arks. They're being used to seek out new worlds and resources to conquer, conquer those worlds, and patrol the Empire's borders against their greedy, imperialist, expansionist neighbors, the Romulans and the Federation. And they need to! That's the system, that's how life works in the Empire. The Klingons aren't equipped for sustainable production within their borders, because consuming to excess and then stealing more, expanding ever outward, is their model, so they can't recall those ships indefinately. They can't reduce their border patrols, because years, if not centuries, of arms races with their neighbors have meant a truly prodigious amount of military might on all sides has to go into ensuring there
isn't a war, so if they draw down unilaterally, the Romulans will just sneak in and start cutting off loose bits of territory by bribing local governors and quietly taking sides in internal conflicts, which are about to become a lot more common, and the Federation will start glad-handing and bribing worlds in the Neutral Zone without any Klingons around to keep them on their toes, and the Organians will just start shoveling planets at them since they're being more effectively developed by default.
Then there's the human (well, Klingon) factor. The Empire no longer needs soldiers. Their enemy is not holding a phaser or flying a starship, it's their own sun bleaching their forests and oceans dead. Soldiers can't fight that enemy. They need construction workers to raise new cities off the homeworld, farmers to create new supplies of crops and livestock, doctors to keep the people still on Qo'noS alive, scientists to figure out if its possible to reverse the damage, and engineers to put whatever they figure out into practice, but unfortunately, the Empire's best and brightest have been funneled into being phaser fodder for hundreds of years. They don't have the human (Klingon) capital they need to combat the disaster, and they don't have the time to develop it, even if they could find the inclination to abandon their entire way of life (which, I've heard, is a big ask, even in the face of total environmental catastrophe).
It's not a matter of just taking a dozen people off of the homeworld at a time on each of their Birds of Prey and dropping them on some other planet with a ration pack and a hearty "Qapla'" to get them started on rebuilding civilization.
Now, if only there was a solution, some way to reduce the Empire's need for the military albatross around its neck, to get a leg up on a massive economic and humanitarian mobilization, and to receive the benefit of the most insanely, improbably effective scientific apparatus in galactic history. Oh, wait, there is; the Federation. All they need to swallow their pride, check their preconceptions and prejudices and fears that Federation diplomacy is just their own imperialist expansionism with less honest branding, and make a deal; end the arms race. Promise to stop posturing their fleet on the Federation frontier, if the Federation will demobilize their own fleet from the Klingon border, so they can have all the fun of military deterrence without the ever-inflating scale of it. Then, once that works, you can draw on the Federation's commerce and industrial capabilities to help relocate the population of the homeworld (or, as apparently happened, repair enough damage for Qo'noS to remain habitable well into the next century, at least) beyond what even a no-longer-distracted Klingon Empire could achieve on its own.
