I'm Sink With It (Optimal Butterfly Sinking)

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DISCLAIMER: While this is Mirror Dungeon usable, it is intended for Boss Encounters. In any content that has lots of mooks (like MD), use Index Yi Sang instead of Solemn Lament! Solemn Lament is otherwise better, especially in fights with SP.

First and foremost: EGO slots that aren't filled can be filled with whatever you have and can fuel; I've only added truly "important" ones.

Anyhow, this is what I believe to be the strongest possible Sinking team for Boss Fights. Thanks to recent ID releases and current Boss design, Solemn Lament - by which I mean, Butterfly - has been receiving immense buffs. Prior to them, Solemn Lament was already better in most circumstances, but was slower than Spicebush in some occasions. No longer!

Even though most of it should be obvious, an explanation on why the IDs were chosen, in order of deployment:

[Yi Sang] Lobotomy E.G.O:: Solemn Lament: The core of the team. All his skills inflict The Living/The Departed (Butterfly for short), with the former providing SP healing and Sinking potency at turn end (and becoming Departed), and the latter dealing bonus gloom damage on every hit based on Sinking Potency on the enemy. 15 per hit in non-SP fights, 30 per hit in SP fights - thats a lot of damage!

[Ishmael] Jeong's Office Rep: While there is some aspect of RNG to the card she starts with (Red card being the best), she not only has a count neutral S1 and S3, the latter of which also boosts our damage using Tremor - Decay, but she also has a free bonus attack every couple of turns with up to 5 coins - all count neutral - to proc even more hits of Butterfly. She also has access to the strongest Sinking EGO in the game: Bygone Days. With its corrosion (which you can fire even if it will partially hit your allies, as it will heal them anyway), you get to inflict Echoes of the Manor, a very important status effect considering most of our skills are count negative, even if only barely. Once you have applied Echoes once, your Sinking stack might as well be infinite!

[Outis] LCA Udjat Vanguard Team 3 Leader: "If Echoes is so insane, why isn't Butler here?" The answer to that question is that Udjat Outis does (almost) everything you could ever want Butler for, and more! She's fast, tanky, clashes insanely well, does high damage on her S2/S3, is barely count negative on all her skills, has a unique panic type that gives you +3 Count per turn (Blue Sand), has a permanent semi-fragile for Gloom and Sloth that also does a "force-proc" of sinking once capped (Sheut Fracture), and to top it all off, she inflicts Gloom Resist Down - increasing the damage Butterfly (and Sinking where applicable) does!

[Rodion] Lobotomy E.G.O:: The Sword Sharpened with Tears: You know her, you love her, it's Kodya. Her negative form is insanely Sinking Count positive, on top of dealing lots of damage. She's our primary source of Count outside of Echoes, and is what allows us to stack to ridiculous numbers. And although it comes up rarely, she has access to Rime Shank, which is a viable turn 1 Superbia target for +8 Count.

[Ryōshū] Lobotomy E.G.O:: Faint Aroma & Solitude: As mentioned with Ishmael, Butterflies love having free skills. Well, Faint Aroma Ryoshu has multiple, which can all proc in the same turn. In addition to that, all her main skills (including her main free skill, her S1) are completely count neutral, she has AoE access, does multiple free "procs" of Sinking - unlike Outis, who costs a count - and inflicts a unique panic effect (Solitude) which not only is 1 Gloom Fragility, but also +2 Count every turn. What more could you want? (Notably, she DOES get a lot worse in non-SP fights, but there Sinking already does a lot more damage, so the trade off really isnt anything to write home about.)

[Faust] Wuthering Heights Butler: Echoes is very very important to have inflicted a single time. But you dont want to rely solely on Bygone Days Ishmael for it, and you dont want to swap out Udjat Outis for Butler Outis either. Well, Faust is a pretty decent source of it. You have a 50% chance of seeing her S3 skill by turn 2, just in time for Kodya's rampup. Even if you dont, her S2 is a lust (big!) skill that's count positive. It's not like seeing her S3 later is bad, either - really, her only downside is her rather poor clashing, which is mitigated by the RNG manipulation Sinking has access to (with Panic types, or just regular SP in SP-based fights).

[Gregor] Lobotomy E.G.O::Lamp: Solemn Lament is an insanely strong EGO, especially with Butterfly. You want it on your team. This is the best Gregor for it, and for Butterfly. He has high coin amounts but is barely negative on S2, and is neutral on S3. He retains the Coin Power Up from Edgar Heir, while also supporting the team with both his passive (Guardians of the Forest) and his unique Panic effect ( Dazzle Dazzle), on top of just being very tanky and able to redirect everything.

[Don Quixote] The Index Proxy - Effloresced E.G.O:: Procuration: Her support passive gives 15 SP to our lowest SP non-Kodya Identity, making them ramp a lot quicker. She's also the best tag in if someone dies, like Butler Faust (low-level Echoes pump-and-dump Butler Faust is a valid strat, but its not needed).

[Heathcliff] Wild Hunt: His support passive is up to +2 Count per turn in non-SP fights, and a free damage up in SP fights. Better than the ID, and has been for ages.

[Meursault] The Thumb East Capo IIII: We have (unique) Ammo IDs that go fast, so his support passive is free damage.

[Hong Lu] Full-Stop Office Rep: We have Ammo IDs, so his support passive is free damage.

[Sinclair] Los Mariachis Jefe: Free damage in SP fights.

The deployment order of our first 3 IDs is very important here - not for bonus slots, but for two reasons: SP Tie Breaking for Wild Hunt Heathcliff's support passive, and speed tie breaking. To explain: When an effect checks for the "highest" or "lowest" of something, the IDs deployed earlier will be favored in case of a tie.

In a non-SP fight, Wild Hunt Heathcliff's support passive provides +1 Sinking Count twice per turn when inflicting any Sinking effect (including Butterfly) with the highest SP ally. So, you want to have this effect apply to units who inflict at least 2 instances of any Sinking with every one of their Skills. Solemn Lament Yi Sang is the best for this, as it makes all his skills Count positive, and one of the applications is always before using a Coin. Hanafuda Ish is the next best, as while the double infliction isnt guaranteed, its still consistent enough - which is important, since Yi Sang wont often be the highest SP ally.
Hanafuda Ish is also a better recipient for the passive in SP fights, which helps this choice.

As for the Speed ties: You want Solemn Lament Yi Sang to go as early as possible, as you arent guaranteed to have 100% Butterfly uptime, even with Greg to supplement. By placing him ahead of everyone else, he will win every tie, and can thus even go before Udjat Outis if you're lucky. Speaking of which: Her Min Speed roll is 7 with

The Udjat [Vanguard], which also happens to be the max speed of many of the other IDs. So we place her behind Ishmael - because we dont want her to get the Wild Hunt Heath buff - but ahead of everyone else, so that she'll always outspeed in order to get max value out of Sheut Fracture and the Gloom Resist Down she inflicts.

Your main gameplay loop ends up being prepping in the first few turns, setting up for when Echoes is live to make a nigh-undroppable stack (don't be afraid to use a Defense Skill and losing out on damage if it means not dropping the stack), and then dumping a bunch of high coin amount skills into a set up Butterfly stack. Most ideally, you use Yi Sang's Skill 3 and pop as many coins as possible during that turn or the turn after, though you can't always control it (with regards to Ryoshu, mainly).
Note that for Bosses with health gates and/or multiple phases: Even if they purge their negative effects, the Panic-type changing effects ALL will remain. So you can pop a Faust S3/Bygone Days Ish right before such a Health Gate to make sure your Stack can recover basically immediately. Additionally, Butterflies bypass the reduction to status damage that bosses like Rien have.

In terms of EGO:
Truthfully, the only EGO you need for this team are Bygone Days Ishmael, Solemn Lament Gregor and maaaaybe Rime Shank (its used rarely, but is gamechanging when it is). In fact, in Speed clears, corroding either Rime Shank or Bygone Days turn 1 with Superbia is quite common. The latter especially allows for 2 turns of Bygone Days in a row, if starting at 0 SP.

The other EGO are just generally quite good and do strengthen the setup noticably, but arent mandatory at all.
Obviously it is sadly no longer available, but Tidal Elegy makes Hanafuda Ish a LOT stronger. Its the best HE Ego for her, but Christmas Nightmare is also notable for making a yellow card start not suck. Hundred-Footed Death Maggot is also extremely spammable due to its incredibly low resource costs, which Hanafuda Ish also loves, so I do recommend it quite a lot.
Honestly, if it wasnt for how Butterflies work, I'd say Yi Sang's Bygone Days is a mandatory EGO. It's passive is very strong, and it's synergy with Echoes is INSANE, as contrary to the description, it actually inflicts a bunch of individual applications of 1 Sinking, so it's essentially a Rime Shank. Unfortunately, using an EGO often means dropping your Butterfly Stack unless you got Greg as backup, so you cant do it that much.

EDIT: Here is my current Sinking ID tier list. If you have any questions regarding a specific IDs placement on the tier list, feel free to ask. (Aside from the top tier, the tiers are unordered)

Lastly, if you wanna see the closest thing to the "peak" of what the team can do, I've attached my own 7-turn Rien clear (counting the final forced turn) to this guide. Obviously, it's missing Greg, and it's not how you would play it in "proper" gameplay as it is solely for getting a fast clear with the lowest effort possible, but I still believe it helpful for understanding how the team rolls.

I'm Sink With It (Optimal Butterfly Sinking)

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DISCLAIMER: While this is Mirror Dungeon usable, it is intended for Boss Encounters. In any content that has lots of mooks (like MD), use Index Yi Sang instead of Solemn Lament! Solemn Lament is otherwise better, especially in fights with SP.