Building L!Ryoma

BRAVE SWORD QUAD.
No, seriously, with flier skills and the boon I think you can actually pull that off. Slap a DB4 for extra 32 damage.

Firesweep(or FS+), moonbow db3(or4) and buishido.
Theres your budget build, it just helps dish out plenty of damage without interruption

What I went for (or at least trying to go for) is Brazen Atk/Spd 3 + Vantage 3. Maybe slap Atk/Res 3 seal on S slot just for the fun of it. Keeping Raijinto because of DC.

He is tricky since he doesn't have access to infantry skills like Steady Breath which make his original version one of my main units.

I got a -hp +res version of him, but what I'm doing could work for you. I wanted him to stay a fast distant counter unit with mixed bulk and self sustain that could also duel melee sword units AND do so on a budget. XD

Raijinto
Reposition (optional)
Aether (Glimmer)
Sturdy Stance 2
Bushido (Sword Breaker 3)
Goad Fliers (optional)
Atk/Def Bond 3

So far Atk/Def Bond + Sturdy Stance 2 have helped raise his physical bulk to respective levels. I got the seal from the last Tempest and Sturdy Stance from a Grail Shop copy of H! Dorcas.

I also have a unit buff him with Fortify Fliers every turn and just let him bait enemies. I usually have Bridal Ninian buffing him because her Fresh Flowers also grants L! Ryoma +4 spd every turn he starts next to her. I also put Bow Hinoka on the team thanks to her weapon's drive atk effect. The last spot on the team I switch between Myrrh (Iote's Shield) and Hector (ally support S rank)

Basically, L! Ryoma is still waiting for the right combination of skills that could make him as independent a fighter as original Ryoma. They are both still great units, but L! Ryoma needs a little more support from his team to be that map clearing monster.

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Sorry I got off on a tangent there, but to focus on your other question for other swords for him, well you ruled out Firesweep (which I don't like either) so that leaves Wo Dao, Slaying Edge, or Armorsmasher. And I can easily beat builds like those when I come across them in arena.

Though Wo Dao + Bushido might be interesting with Glimmer for + 20 Damage every special proc. And pair it with a good Brazen skill, but it does want Desperation as well, which doesn't have a seal yet, so maybe Brash Assault or Vantage? Could also go complete glass cannon by adding Life and Death as well instead of a Brazen skill.

As for quad builds, sadly I don't find them as amazing as they sound. Your quad L! Ryoma would have similar atk, spd, and lowish def as Elinica. With her I have found that she needs to be able to get the job done with the first double strike or she might not live long enough to get in the 3rd and 4th hits. It's a common problem when I have her attack an enemy M Grima/Green Armors/Dragons. To solve that I also buff her with Fortify Fliers on Tana. Death Blow 4 would be nice, but is an unrealistically ultra premium skill at the moment. Maybe after IS finally releases that Skill Refine shop everyone wants.

Anyway, hope you can find a fun build you like. And remember that the other team units will really help your build.

I don't even like the firesweep build on him. I find it lame when he can actually take a hit if you know what you're doing, plus there's something about this big guy going for firesweep that just seems cowardly.

There's slaying edge desperation galeforce, or here's an alternative I've been promoting (or a variation that works with your IV, anyways). Once he takes a hit, he goes up to 35/34 defenses, and he'll heal a bit of HP every round.