Long before I posted this topic, I got to the 4th stratum in Etrian Odyssey 3 with a solo Wildling. This afternoon, I decided to try to finish it at last.
The most viable classes for solo runs in this EO are Hoplite, Wildling, and Yggdroid.
Hoplites are my favorite defensive character in this series because they have a passive skill that heals them when they defend for 60 HP. The closest equivalent to that move is a Dragoon skill in EO5, but it's an active skill that uses TP and only works when the user takes damage. Hoplites use spears like the name would suggest, unlike the inaccurate blunt weapons of EO4 Fortresses. Spears let them hit from the back row without penalty too. Why is that important for a solo character?
Once you get to the Deep City after the end of the 2nd stratum, your characters can get a subclass, which does not have any restrictions except for being unable to use the subclass's "class passive skill". For solo characters, this means being able to summon clones in battle at the cost of splitting HP and TP between them with the Ninja's Bunshin skill. Add in Tagen Battou, which sacrifices the clones to unleash the strongest attack in the game. Some of the later bosses in both the Armoroad and Deep City stories heal themselves, so you need Tagen Battou if you're not using a Wildling or an Yggdroid.
Wildlings are a lot different. They're the status ailment class in the one EO game where they're less effective. However, they're not like EO Untold 2 Hexers where trying to inflict status ailments is the only thing they can do. Wildlings summon animals to fight for them, and they can get passives to make their animals take hits for them. Since this class requires so many skill points, you'll want to focus on two animals. One for random encounters, and one for bosses and FOEs.
It took some experimenting to get it right. Some players love the Tiger, but after its initial instant kill roar, it can only counterattack. This is unreliable without specific setups. I use the Elephant instead. It hit all enemies on the turn when they're summoned, and on later turns they hit single targets. All the Elephant's attacks can cause Panic, which prevents enemies from using any abilities other than their basic attacks, and has a chance of making them hit themselves. For boss fights later in the main game, go with the Lion. It hits much harder than the Elephant, and can stun enemies, which works exactly like Flinching in Pokemon. The problem is that the Lion falls asleep after every attack. This is why my Wildling has a Monk subclass. The Monk's Refresh skill can wake up the Lion every turn for a low TP cost. No having to fill up your limited item slots with Theriaca B required. (No, I'm not the first solo Wildling player, because there's a PKLooove on YouTube who did it better. He likes the Tiger a lot more than I do.)
I don't have much experience with Yggdroids, but they have a move called Shoot that is similar to Tagen Battou, and they're immune to binds that aren't self-inflicted with their gimmick skills. Since you get that class so late on the Deep City route, you'll have to use a New Game Plus file if you want to play the entire game with one.