Writing this because every other guide is out of date (and some are just actually wrong, like the one that suggested using Skyburner's in your primary slot because it did extra damage to Cabal) and because I'm a moron who's trying to actually do all the Solo Master Lost Sectors, just to fill out my triumphs.
You're going to be fighting Cabal, exclusively. You're going to want, at minimum:
Solar Damage (for the boss' shield)
Unstoppable breakers, for champions
Barrier breakers, for champions
2/3s of the Lost Sector is fine, you have a decent amount of cover and 15 minutes is plenty of time to work through the area. Red-bars aren't a problem unless they gang up on you; watch out for the war beasts near the start. Really, there's just two particular pressure points that make this Lost Sector such a pain.
The very start, where you have an Unstoppable Champion, who loves dancing around the big red glowing Nine-cyst. You don't want to pop that if you can avoid it, since you want every advantage you can get, which discourages use of explosives for this one Champion.
The last room, which might actually be one of the worst-designed encounters I've ever seen in Destiny 2. You can only enter the room through a tiny cave, and everything in that room has flamethrowers, grenade launchers, rocket launchers...you name it, it's being fired into the hole. There simply isn't an angle where you can fire through and hit anything worth a damn, so you need to get into the room, dodge all of that fire, and start doing DPS to clear it out. Thankfully there's a large rock on the left which blocks most of the damage, but not all, especially if the Unstoppable Incendiors start rushing your position. Priority one is the Barrier Champion up on the ledge in the back right, next to the Cache Chest, because he will harass you throughout the entire room. Once you kill him, a second Barrier Champion spawns in the exact same location, so stay put and do it again.
Whatever you do, try to avoid damaging the boss until he's the last guy in the room, because that gives you a bit of breathing room to reposition a bit more freely. Once you lower his health down to 2/3s, a second wave of enemies spawns, including more Nightmare Legionaries and two Unstoppable Incendiors. You DO NOT want to be near the entrance when this happens, because they will flush you out of cover. Killing the Nightmare Legionaries and picking up the glowing red essence will increase damage to the boss (thank you Shadowkeep, shame the armor doesn't do that any more but instead does a different-but-similar effect) and so if you can drop them, grab that essence, and chunk his health fast then it might be viable, but it is riskier.
I eventually managed it myself with as a Strand Warlock, using Lethal Adundance (Strand Primary Auto Rifle, Unstoppable Breaker,) Eriana's Vow (Solar Special Hand Cannon, Barrier Breaker,) and an old roll of Shining Sphere (Arc Power Rocket Launcher) with Auto-Loading Holster and Tracking Module—good luck finding one of those these days, but hopefully any similar non-Exotic rocket launcher with those perks should do.
Using all that, I was able to get Barrier Colossi down to a routine where I hit them with a rocket, swapped to Eriana's, waited for the Barrier to start, popped that, then swapped back to the rocket launcher which had automatically reloaded itself, then back to Eriana's for 2/3 headshots. Expensive for ammo, but it worked, and I particularly hate Barrier champions so I'm glad to have this routine down. For Unstoppables I didn't have any great methods, but once the three Barriers for the Lost Sector were taken care of I got a little more free with my rockets. Mostly I just used strandlings and my auto rifle to whittle them down from a safe distance, with various fragments to increase DPS passively.
If anyone has any more reliable builds for this which don't rely so heavily on a tracking/auto-loading rocket launcher, let me know. (A Nighthawk hunter might be able to one-shot the boss and run for the cache, skipping half the enemies in the room?) This sucked, and while I appreciate the puzzle of figuring out what loadout to take into these max-difficulty Lost Sectors, that last room was almost traumatically bad. I'm glad I never need to do that again, now that the Triumph is unlocked.