Some general comments... Be aware that the strategies below can overlap. Alice's Marionette Pareil, for example, guards well against most of the bosses' deadly fullscreens. And both Akame and Chihiro have big elemental weaknesses, so Nitori with her elemental shells will do big damage, as will Marisa's appropriate elemental spells, or Sakuya or Youmu when elemental weapon-ed with Byakuren, and Sakuya, while she's at it, can easily Private Square them dead with the appropriate buffs! Each boss has a variety of ways in which they can be killed, and which way you use is entirely up to you.
The annoying thing about this stage, however, is that the bosses must each be beaten with a different team - and each one of them in turn. Sure, you can assemble a team of Byaku, Youmu, Sakuya, Nitori, Alice and curb-stop one boss into oblivion; but what will you then do about the other two? Getting this balance right is the first key to success in this stage. The creeps each route throws at you to give some indication on what is required, but don't be afraid to rejiggle your teams before the boss fight as you see fit.
And as if that wasn't enough, each boss has ridulously powerful "hax attacks" that you seem to have have virtually no way defending against regardless of who you bring. As such, like many other things in this otherwise fantastic RPG, success in this stage depends on plain old luck. Certainly do not expect success immediately, but try to get your strategy set-up, and if you seem to be doing pristine with it until you get screwed over in one turn, its probably viable. Reset and try again - and pray that on at least one single occassion, luck remains on your side for all 3 battles!
Middle Route (1st Team):
Team: Mokou, Patchouli, Nitori, Reimu, Youmu, Aya as commander.
あかめ (Akame) has 20,000 HP. She is weak to fire-type moves. She resists Ice, Light, and Dark attacks. She has 300 Defense and 150 Magic Defense.
If you have the formation you can obtain by talking to Orin, use it. It boosts fire-related parameters.
First turn, Akame will use a poison-related field type move which will boost the chance of poison moves activating and severely increase poison damage. You'll want to get rid of this as soon as possible. Patchouli has a field-type move that manipulates for Fire(ホットスポットヴォルカーノ, Hot Spot Vulcan) , so it's recommended to use that on the first turn to cancel out the poison field. If your Patchouli somehow goes before Akame does, then have your Patchouli use it on the second turn. Nitori should have 20 points in her Gun tree, and you should spend the first turn using Elemental Shells (her 5point Gun move) to enchant her attacks with Fire, and then spending every turn after that using Spitfire (her 13point Gun attack), reapplying Elemental Shells as needed. (N/B: Alternatively, counter the poison field with Sanae's null-poison field, obtained after 13 points on her snake skill tree.)
Youmu should be running her Human Realm Sword stance, and should be running Katanas as well. You can go with 20 in both trees, but 25 in both is preferred, for increased critical hit chance (25 point Katana), and for increased duration of the stance (25 point HRS Stance). Her attack of choice should be Slash of the Eternity (未来永劫斬, level 32), with Crescent Moon Slash (弦月斬) being used during cooldowns.
Mokou should spend the first few turns using Tsuki no Iwakasa's Curse (月のいはかさの呪い), as Akame can be inflicted with the Curse status, and this should help keep her healing in check.
Reimu is there to heal status ailments and general HP recovery, and because she doesn't fit in anywhere else.
After the first few turns, Patchouli should be using Fire attacks as well, reapplying Hot Spot Volcan as needed to counter the poison field. Patchouli, Nitori, and Youmu will carry you in this fight, barring you don't get screwed over by unlucky fullscreens - there's not much you can do about them without Alice, but Alice is being used elsewhere.
Addendum: Reimu at level 51 gets a fullscreen barrier move (二重大結界, Super Duplex Barrier) that you can use to protect your party against the fullscreens mentioned above, but, as there is no charge-up or indication of when Akame will use the fullscreens, predicting that will be very hard, and thus is not recommended.
As mentioned above the private square tactic works well against her. With her latest 2-handed sword and hen buffed with Fire weapon and weapon bless, she did 1200 x 3 a pop; double that if on last word. Get Byakuren to cast them - with weapon bless first and fire weapon after; then Private Square. Recast fire weapon when Byakuren moves again. These should be the only 2 turns the boss gets - just hope they don't kill either Byakuren or Sakuya in this time!
Left Route (at the start, the right-most door at the end when entering the boss room) (2nd Team):
Team: Sakuya, Byakuren, Satori
Before attempting this fight, make sure your Sakuya has decent-to-high resistance for Blind.
千尋 (Chihiro) has 26,000 HP. She is weak to Lightning attacks. She is immune to Ice, and resists Light, Dark, and Blunt/Crush attacks. She has 150 Defense and 50 Magic Defense.
Sakuya should be using Private Square (プライベートスクウェア) with 25 points in her Time Stop tree. This allows her Private Squares to last three turns instead of two, cost 25 MP instead of 33 MP, and basically, allows her to keep bosses locked in permanent "stun." Her attack of choice inside Private Square should be Full Moon Revenge (フルムーンリベンジ, 25 point Great Sword/Two-Handed Sword tree). Byakuren should be using Elemental Weapon (エレメンタルウェポン), Weapon Bless (ウエポンブレス), and Dark Heal (ダークヒール). Byakuren should have 25 points in her Support Magic tree. Satori should be using Soften (軟化術) with 25 points in her Id tree. Make sure Satori has all of your +IND accessories.
When this fight starts, Sakuya should immediately Private Square, Byakuren does a Lightning-property Elemental Weapon on Sakuya, and Satori uses Soften on the boss. If Soften does not debuff the boss's defense, then F12 and restart the fight - not having that Defense debuff is crippling.
Once Sakuya's Private Square lands, use Full Moon Revenge every single turn (unless Sakuya is favored by the RNG and gets a Last Word, in which case immediately stop and use it), but always making sure you have 25 MP available to Private Square once the current one resolves. Reapply buffs as needed, and never let your Elemental Weapon wear off - keep in mind that Sakuya's Private Square is three turns, so if, at the time of casting Private Square, she only has three turns left on the buff, it will resolve after her second turn inside Private Square. Ideally, when casting Private Square, you should have atleast four turns left on your Elemental Weapon buff - as a result, it'll be very hard to find a spot to cast Weapon Bless on Sakuya. Fortunately, Sakuya can easily carry the fight with Elemental Weapon and Satori's debuffs, so Weapon Bless is not necessarily needed.
Alternatively, grab a team that includes Reimu, Patchouli, and an elemental attacker (Nitori or Byaku+youmu/sakuya). As mentioned, give your physical attacker blind resistance. Also, patchy also has an attack that increases lightning attacks (heaven crack, ヘブンクラック): use it! She can be dealing 3000+ damage a pop with it. Also pay due attention to her hax attacks listed below:
-オクトパス・メイルシュトローム: she does this at the start and subsequently after every charge up. Damage can be nulled by Reimu's water barrier. For its instant KO to not activate, land power must not be in favour of water when cast, so manipulate it otherwise with Patchouli. Hope also that she doesn't unleash it again before Reimu's barrier spell cools down: the obscene damage it deals means almost certainly a game-over. - アシッドストーム: survivable on its own; but if cast twice in a row, expect alot of deaths.. -触手, also 触手乱舞: almost certainly a OHKO, even to high PDEF characters. Crush element (殴), so the relevant damage-reducing accessories can help make it survivable.
Right Route (at the start, the left-most door at the end when entering the boss room) (3rd Team)
Team: Marisa, Sanae, Alice
カガミ (Kagami) has 23,000 HP. She is not weak to any kind of attack, but she does resist Sword, Pierce, and Blunt/Crush attacks. She has 600 Defense and 0 Magic Defense. That is not an error.
Marisa should be using Rapidfire Master Spark (速射型マスタースパーク, level 55) and Master Spark (マスタースパーク). Galaxy (ギャラクシー, level 51) will suffice if your Marisa does not have Rapidfire Master Spark. Marisa should be using 25 points in her Master Spark tree and her Magic Mushrooms tree. Alice should be using Marionette Pariel (マリオネットパラル) and Little Legion (リトルレギオン, level 28). Sanae should be using Prayer's Wind (祈りの風), Shield's Wind (盾の風), Super Healing Miracle (超回復の奇跡, level 40), Star Ritual to Call the Godly Winds (神風を喚ぶ星の儀式, level 36), and Miraculous Fortune (奇跡のおみくじ, level 24). The formation should be anything that places everyone in a straight line. If you have it, equip your Marisa with the Magical Booster (マジカルブースター) rod. It can be obtained as an item drop from the Kraken Magician enemies in stage 5. Your Alice should be equipped with an item that gives her a circle-within-a-circle (◎) resistance to Dark.
Marisa's turns will be spent alternating Rapidfire Master Spark and Master Spark. Alice's turns will be spent using Marionette Pariel, and using Little Legion the turn immediately after Kagami does a charge-up.
Sanae should use her defense buff the first turn, the magic attack buff the second turn, and the Maximum HP buff the third turn. Prioritize healing over the buffs, however, if someone is low on health. Revive anyone as needed, reapply buffs as needed, and keep everyone's HP at high amounts, and you should be good to go.
Kagami has a buff (エレメンタルバリア) that raises her Fire, Ice, Lightning, and Earth resistances drastically. Ignore this, as it has no impact on the Mystic-based Marisa strategy.
(Regarding the buff Kagami cast, it is a variety-type spell, therefore it can be dispelled using Sanae's anti-buff/debuff (the blue orb icon) skill. It is not required for Marisa to invest on the Master Spark skill tree. Bomb charges from enemies can be dispelled as well. However, success to cast depends on enemies' RES and caster's IND. )