1. Landmarks.
You start the level looking at where Karn emerges. This is what I'll call the Pit. To your right is a structure. This is called the Right Structure. The opposite side is called the Left Structure. Remember these terms.
2. Weapons.
I carried a Torque Bow into this level from the previous. It's not essential, but I found it more helpful than a Gnasher would be. Know the layout of where weapons are.
There's a Markza to the left of the Pit, by an infinite ammo box, resting on a memorial. There's a Longshot by the right of where you start, along with a Boomshot and an infinite ammo box. There is also a Onyx Guard Ammo Case, in a wall crack, just to the right of the Pit.
Each of these weapon spots also have Frag Grenades at them. 3 spots with 1 each, giving you a potential of 6 grenades. There's also 1 grenade from the Onyx Ammo Case, but lets forget about that for now.
3. The Plan
Level starts, and you activate the declassified option. Don't worry about grabbing anything, just keep a focus on Karn.
If you haven't noticed after playing several times, Karn goes up each Structure once. He never just goes up one. This is important. I always made sure he'd go up the Right Structure. I'll explain later. When you finally destroy his flame grenade launcher, he'll go up a Structure. You want to make sure he goes up the Right one, so only "Wounded" him when he is close by to it. If he goes up the Left Structure, you're on you're own from here.
When he begins to head towards the Right Structure, stay in cover. Always be opposite the side he is. I would hide behind a full cover memorial, a few blocks to the left of where you spawn. It's the one with a tiny blue ammo box beside it. You want to be here because his support teams will spawn near where he is. If he's left, they'll spawn more to the left side. If he's right, there'll be more spawned at the right side.
His Shibboleth will roar, and the Mauler, Therons and Drones will appear. Focus on shooting the Therons, as their Torque Bows will instantly kill you. Don't worry if you don't kill all of them, just try to get at least 1 or 2 killed. Always spot them too. If your teammates aren't killing them, at least you'll briefly know where they are. One enemy should usually appear just to your left, so try to take them out first for an easy kill, regardless of them being a Theron or not.
***Also, and this is key too, don't stop looking at Karn. Sometimes, and I do mean sometimes, as it's inconsistent, and I only had it happen a few times over the hours I played, he'll leap at you. If this happens, you're practically dead. There really is no warning animation. No teammates say anything. It will just be a moment, and then he and his pet are destroying your cover and roaring in your face.***
When Karn comes down, wait for him to charge, and them move about a 1/4 way around the level in a clockwise direction. Always stay outside, in the gray floored area. The enemy will usually stay in the middle, and the distance will help you avoid being shot by them. Don't shoot at Karn. Shoot at the enemies, spotting them constantly.
Continue waiting for Karn to charge, moving 1/4 way around in a consistent clockwise direction, getting behind cover, and shooting the remaining enemies.
Wait until all of them are dead to begin you next plan. Keep an eye on the Mauler, because you'll need to make a judgement call here. I had been using the Boomshield he drops to fight Karn for most of my attempts. I would grab it AND stay behind cover. I would use the Markza with the Boomshield, blasting the heads off of Ragers the best I could, but it usually wouldn't work well. You can't shoot over cover, even though your crosshairs say you can. You'll have to go to the sides of the memorial covers if you want to do this.
If you use the Boomshield, it's great for battling Karn because his turret fire will almost always be blocked by the shield. You're just unable to fire as easily...in a way.
The time I finally beat him, I swapped out my Torque Bow for the Boomshot, while also keeping my Lancer. The reason is because the Markza, while great for headshots, doesn't fire fast enough for a later part.
Anyway, decide what weapon you think is best. Keep avoiding Karn, move around, grab the Grenades from the Longshot/Boomshot pickup area and the Markza pickup area. Don't grab the grenades by the Onyx Case. Just don't.
Once you feel like you've collected enough, lead Karn around to the left side, and rush around to the Onyx Ammo Case area, taking cover by the Pit. You want to be near it, but not "in it" with your cover. Don't be touching the curved gravel area, be at cover between the Pit and the Right structure, just slightly closer to the Pit side. By now you should have a feel for how many shots it takes to get Karn Wounded, but, if you aren't, it usually takes 2-3 sequences of shooting the weak spots of his pet.
Just try to make sure you're on the Right side of the level, and Karn is to the Left side of the level. If you don't do that, and destroy his leg, he'll fire on you the entire time he crosses over to the opposite side of the map. This would be deadly if you're on the Left side and he has to cross to the Left Structure (unless you have the Boomshield). Just try to be near the Onyx Ammo Case, and have him near the Left Structure.
This is when he launches 4 Nemacyst at you. Just move left and then back to right. They tend not to really bother you, and usually head in one direction. If they hit you, though, it should be fatal, so avoid them. Just look up, and move the opposite way they're heading. Then move back to the cover by the Pit and the Onyx Ammo Case that you were just at.
If you're using the Markza with the Boomshot, good luck here. Make sure you try to get headshots on both the Ragers and the Grinders. Grinders are very dangerous, even at range, because of how strong their shots are. Ragers will go for headshots, but with the boomshield, try to kill from your right to your left. Remember, you're most exposed to the right side of your Boomshield, so doing it this way will help you clear them out in a safer setup.
If you're holding a Lancer and Boomshot like I did, just stay in cover. Let the Ragers and Grinders slowly move to you. To your left there should be a lone Grinder. When you can lean out left safely, spot him and shoot him with your Lancer. When he dies, Karn should be coming down. Stay by the Pit. This is key.
At this point Karn swings wide right and left. He approaches you, and tries to force you have to move around your cover. This is so you get up and are either shot by Grinders, Ragers or Karn, or are ripped apart by a charging Rager. By staying close to the Pit, though, you prevent this. The Pit is essentially an invisible wall that neither you nor her can pass. So he'll try to swing wide to your right, but can't. In cover his shots will almost look like they're about to or are hitting you, but you'll be safe. He'll then try to swing wide to your left, but the circular shape of the map, and the way the memorial covers all face will allow you to not need to move. This means he's close enough to accurately blind fire at with your Lancer. Fire blindly at him, and nail those active reloads.
In the meantime the Ragers should be approaching. Because you're near the invisible wall, they should cluster towards you. If they're in a big enough group, blind fire your frags at them. Use at least two to prevent their Rage attack. If they become enraged, quickly fire your Boomshot at them. Don't waste the Boomshots on Karn, as it travels to slow to really hit him. Use them on the Ragers.
When Karn's beast opens its mouth to launch that explosive spit at you, blind fire at the glowing mouth until it spits. Immediately roll left (or right if you feel you're left side is to dangerous from leftover Ragers or Grinders). The spit tracks you, so you don't want to leave too early or too late. Just let it fire, then roll and roadie run to to the next piece of cover two spots away. Keep blind firing at Karn, let him have his beast spit again, and unload blindly unload on it's mouth.
Do this until it collapses. At this point, fire everything you have left at him If there is a Rager or Grinder remaining, you might want to take them out, just so you don't die at the end. Be careful about peaking out of cover, as Karn is shooting a Boltok wildly at you, and two shots from it will kill you if they hit you.
If you run out of ammo like I did, this is when you want to grab that Onyx Ammo Case. It'll give you a grenade and fill you ammo for the Lancer and Boomshot, so you can blast his ass to hell.
If this works, awesome. You just beat a pretty difficult boss. If it didn't work, review what I wrote, see what didn't work, and try to make up your own changes if you feel this strategy was ineffective.
After hours of play, this probably the safest way of taking him down, as when I took cover by the Pit my teammates were nearby, so they were helping to fire at the approaching enemies, forcing them to cover.
Hopefully this works for you.
And don't forget I did this solo. I have no idea if this would work with other players. Other people are a liability, so you're better off alone on this one.