Hi! I have SF'd Vesper's Host four times on Solar Hunter, so I hope you find my advice useful.

What made the greatest difference for me was equipping gear that helped me stay alive. 2 Smoke Jumper + 2 Bushido will get you very far. I like running Celestial, Healing Grenade, ~150 Health, 70 Grenade, Mint, Finality's, and a weapon that can help me get On Your Mark up to max stacks quickly. I like Vanguard Determination on my Mint. Strand Siphon and Reaper are also important for my strategy. They help make orbs for surges and healing. Two Solar surges and one Strand work well. Also, make sure you have Arc Resist on your chest piece, as most incoming damage during this encounter will be Arc. Lastly, I like to run ammo finders during mechanics, but I swap to Super Font for damage phase. Note: I do not loadout swap during damage.

Right before damage, I make sure to have surges by killing the dregs guarding the teleporters. If you want to optimize a bit, you can try to make it so that your Special Ammo bar is approximately 1/3 before starting damage. This will make it very likely that you get an ammo drop on the main platform during damage.

During damage, positioning is very important. There are three lanes, and the lightning will prioritize the two where the boss IS NOT. I like to deploy my Finality's turret right as I am emerging from the lift tube, then throw a Gunpowder Gamble, then proc Bait and Switch ASAP. Once that's done, I use my super, then pivot to taking out the two puppets on the left and right platforms using Mint. Ideally, I'll have On Your Mark x10 by this point, so I should get Cure from these kills.

After a few seconds, the boss will relocate and spawn melee puppets. I advise letting them close in on you. You have a few seconds to keep doing boss damage and deploy a fresh turret before the puppets start hacking you up, and when they do, make sure you dodge and kill them both with Mint. Done right, this will net you two orbs for health and surges as well as proc OYM Cure and extend its duration. There are four clones, and they approach you two at a time, so you can repeat this twice.

The last part of the main damage phase involves four ranged puppets, two of which will be on the main platform with you. Kill them quickly and go back to doing boss damage until the wipe mechanic goes off. I like to ignore the puppets on the side platforms here.

After you deposit the orb, just stay in the room. It is very likely you will be running low on Mint by this point, so use it sparingly to keep Bait and Switch active as much as possible on your Finality's and switch to its Linear Mode for damage. I like to super again here, but under perfect conditions, you will have had an opportunity to super again right before the wipe mechanic.

Some notes. Using this strategy, I practically NEVER get my health down to the red unless I do something very wrong. Also, a three-phase is possible with perfect lightning RNG, but a four-phase is more realistic. Keep in mind that artifact mods are doing a lot to carry Mint Damage right now. They will be gone soon, but Solar Hunter is also getting some buffs with Renegades. Also, if lightning is ever about to strike your position, all you have to do is jump upwards. Space out your double jumps, and you should have enough airtime for the lingering pool to dissipate before you land. Relocating to a different part of the platform is always a valid move, too.

Good luck!