I'll let someone else speak to Shoguns and Imperials, as I have very little experience with them - I know Shoguns can dual-wield weapons, and Imperials are supposed to be good in protracted fights against bosses. Oh, and there's Highlanders too, who use HP to power a lot of their attacks... might be best when paired with something that can keep HP topped up at all times.
I'm probably missing other classes that could serve as damage dealers here. Geez, this game has a lot of classes.
If there's one kind of class that's definitely over-represented in this game, it's definitely the "Front row glass cannon." You can tell they picked the classes based on popularity, because there sure aren't that many pure tanks or elemental damage dealers. To this game's credit, they all seem pretty useful and fairly distinct from one another and I think you'd be good with any of them SpaceBridge.
Imperials are extremely good boss killers once they get access to the second tier of skills, but until then they basically run out of TP after one or two Assault Drives and can't do a whole lot in random encounters. Let me tell you, as someone who really liked the class in Etrian Odyssey IV, it sure is a different experience bringing one up from level 1 instead of level 40. You also take double damage while prepping a drive skill, which makes them extremely vulnerable to one-shots if you don't have someone tanking or drawing aggro. Now that I've reached the later levels, the Imperial on my team has been pulling her weight, but it was rough going for those first few labyrinths.
I haven't messed around with the Shogun, so I can't speak from personal experience, but it seems like their skillset is a little more esoteric than some of the other frontliners, with a lot of abilities focused around commanding other party members to do things or snowballing damage output as long as they stay alive (while also having skills that make them take more damage or draw aggro.) They seem fun, so I might eventually sub one in for my Farmer if I get tired of a character not hitting things.
I might also make the potential case for a Nightseeker. They overlap with the Arcanist in terms of inflicting ailments, but if you focused on the skills that do more damage to enemies who already have ailments, you could probably get some pretty good offensive synergy going. Of course, you can always have them as a subclass on your frontliners too. Even at half effectiveness, some of those skills seem quite useful.