This is probably the first true test of your party's flexibility in Metaphor: ReFantazio, as all of these bosses require different archetypes to be levelled up in order to defeat them. There are four mini-bosses you're going to have to fight during this quest, but you only actually have to fight three of them to complete it. If you find whichever boss you've opted to fight second truly insurmountable, then you can simply challenge whichever one you didn't challenge instead.
Before attempting to make your way through Louis's Gauntlet Runner, I would recommend you have one level 20 Seeker, two separate Mage/Wizard party members with lightning skills, a decently leveled Magic Knight, and one Gunner with Poison Shot (or alternatively one Poison Stinger item).
Xanth
Xanth is an absolute pushover when burned or poisoned. One party member with the Gunner's Poison Shot skill or the Commander's Fiery Strike skill will get you through this fight on its own, as Xanth will take massive damage every time he attacks on his turn. It'll be more than enough to finish him off before three turns are up.
Alternatively, if you have an item that will inflict poison or burn, like the Poison Stinger consumable, that'll solo this fight by itself.
At this point you can choose to fight either Glechom or Ceiba. Personally, I think Glechom is the easier and most straightforward fight of the two. However, I would suggest backtracking after whichever fight you complete first and fighting the other boss regardless. This is completely optional, but doing so will net you extra gear, gold, and experience.
Glechom
The key to this fight is the Knight Archetype's ability Knight's Proclamation. By using it and making Glechom target one specific unit, he will no longer use his Earsplitting Shriek daze ability or his Punishing Blow multi-targeting attack, and will instead simply use basic attacks. The Knight archetype is also resistant to slash attacks, so the party member that takes aggro will take less damage (as long as you've not attached Knight's Proclamation to a non-Knight party member with Skill Inheritance).
You'll only have two opportunities to use Knight's Proclamation, because using it on the third turn is pointless as the fight will simply end after this if you don't defeat Glechom in time. Simply attack with that party member on the third turn instead.
Glechom isn't weak to any attacks or elements, but he also isn't strong against any either. If you're struggling to damage him enough before the end of the third turn, consider using some high crit attacks to gain extra turns (such as the Warrior's Critical Strike skill or the Merchant's Gold Attack skill) or some medium-strength spells (such as the Wizard's Botra/Blizza/Kandera skills or the Thief's Mudoon skill).
Alternatively, if you have the Faker levelled up high enough you can use Rakunda to decrease his defence for 3 turns. You can also use a Corrosive Acid item for the same effect (if you have one). Defence lowering skills and items stack if you're really struggling.
Ceiba
At the start of combat, Ceiba will cast Mage Wall. This will make him repel all physical attacks. Although you can attack him with magic skills, he's got a pretty chunky health bar and you'll likely be unable to kill him before the end of the third turn as a result.
The most straightforward way to get around this is to either use the Seeker's Tetrabreak skill or the Commander's Heartbreak Stratagem skill. Both of these negate the Mage Will ability, which will make Ceiba weak against all forms of physical attack. Use units with high-damaging physical skills (like the Magic Knight's Heavy Rush, the Gunner's Poison Shot/Hex Shot, or the Brawler's Skull Cracker) to both deal damage and gain extra turns.
Agility dictates the order in which your party members take their turns, so consider altering your party's equipment or changing to a different archetype with extra points in Agility (such as the Thief) and assigning Tetrabreak/Commander's Heartbreak to them via skill inheritance if a party member is too slow.
Of course, if you don't have either the Seeker or Commander at level 20, you won't have the necessary skill to bypass Ceiba's Mage Wall. But worry not, as there are a couple of other, sillier ways to finish this fight with ease.
If you have the Magic Knight and Gunner classes unlocked, this fight can be trivialised with the Magic Knight's Hammer synthesis. Simply send one Gunner and one Magic Knight into the fight, and use the Magic Knight's Hammer synthesis on the Magic Knight's turn as many times as possible.
The first one won't deal much damage, but the other two will likely deal upwards of 1,500–2000 damage (depending on your stats/level/equipment etcetera) as the attack makes the opponent weak to electric once used. Skip turns so that the Magic Knight has extra turns to use on the synthesis, and then guard or use magic attacks afterwards to waste turns so that the Magic Knight will be up next after Ceiba. Using this move three times will guarantee victory.
Alternatively, you can use an even dumber strategy—as long as you don't mind quite literally paying to win. Unlock the Merchant class with all four party members, and then spam Gold Attack with them until Ceiba dies. Gold Attack has a high crit rate and the Merchant has a high base Luck stat, so you'll likely earn one or two criticals and earn yourself some extra turns before every one of Ceiba's turns.
Although this method will require some real-life luck, you can offset this by giving your party Luck-increasing gear. You could also respec your entire party into Brawlers, and unlock the level three Diligent Discipline passive ability. With four Brawlers, this will passively increase each party member's critical hit rate by a combined 20%. Pair this with the Gold Attack ability via Skill Inheritance and absolutely go to town. This will cost you a lot of MAG and money, but I am not exaggerating when I say that you will crit roughly 50% of the time with this ridiculous strategy.
Cabio and Salva
You'll have to take both of these enemies on together, but don't worry—this fight is way more straightforward than the last two.
Salva is weak to piercing, so projectiles and spears are the most effective. Cabio is weak to lightning. I would recommend a mix of Mage/Wizards (for lightning attacks) and at least one Gunner, and one Knight/Magic Knight (for piercing).
However, if you don't have enough points in either Mage or Wizard to unlock Kande/Kandera, you can just as easily win with a bit of brute force. In this case, go in with a full team of Gunners/Knights and at least one Magic Knight.
I'd recommend taking out Salva first, as his Mage almighty ability will target all party members and ignore their elemental and physical resistances. Any Knight and Gunner attacks will take him down swiftly, but use the Magic Knight's Heavy Rush and the Gunner's Poison Shot/Hex Shot for the quickest results.
If you have lightning skills, Cabio is very simple to beat. Spam Kande/Kandera until he dies. If only a few of your units have lighting attacks, skip the turns of the units that don't.
If you don't have any lightning skills, then bring a Gunner and one the Magic Knight. Use the Magic Knight's Hammer synthesis on Cabio to shave off half of his health bar in a single attack.
If you don't have any lightning skills and you don't have the Magic Knight unlocked, go in with three Gunners and one Thief/Merchant. This is admittedly risky (and requires a lot of MP), as Gunners are weak to Cabio's slash attacks, so this method may require a few attempts (during which you'll just have to hope Cabio targets your Thief/Merchant on the first turn).
Respec the party member with the lowest Agility into a Merchant/Thief, as you'll want them to come last in the turn order. Use the Mania Bullet synthesis on Salva three times in a row with your Gunners. He should almost, if not definitely, be dead. For the next two turns, use Mania Bullet on Cabio. Every second time you use it, use it on him—he'll be weak to the attack. Four of these, alongside the extra turns you'll gain from every second Mania Bullet, should finish him off too.