It might work a bit better if you can separate a city that can be accessed by diving that far underwater from a city built that far down underwater. If you had a city built deep underground, that somehow managed to tap into a network of deep caverns with water access (without compromising them), you might end up with a city that builds into and uses those caverns, and ends up with a stable access point where your diving hero can find it. It will also be a lot more plausible if you can accept a shallower city, after all finding the underwater access point to an amphibious city (built partly underwater) can be pretty striking even if it's not quite so far down.
I recall a story where a city was built into a dry portion of a cenote, a cave system with a deep underground water access. Usually the cenote will be filled to the ground water level, so not particularly useful for dry caves deeply underwater (especially not with surface access, which your city would also need to survive), but if you assume a tricky geology and a number of features your city happens to be able to take advantage of (rather than deliberately causing, which they wouldn't have the technology to do yet), it might not be quite so impossible.
So, maybe you start with a deep cavern system filled with water. It needs to have deep ocean access - maybe the system is through a cliff next to the sea (which drops all the way down to give an opening at the required depth). Or perhaps it is an island over an old, inactive ocean volcano, where some of the lava channels are waterproof stay dry (and some have access to the island top and therefore air), even though most of the mountain is under water, and there are systems with access to water. Freshwater might occur in caverns without direct access to the sea by filtering in through the ground rock (aided by pressure and osmosis) into parts of the cavern system where there isn't a direct connection to the sea. In other places, sea-water would have direct access to the cavern system, and maybe there would be some brackish water tunnels where they systems interconnect.
So, parts of the system get cut off from each other and the ultimate water source (caverns that hold water, should be water proof enough to keep water out if that initial access is blocked) perhaps through earthquakes. There ends up being dry caverns extending deeply under sea-level, in addition to the fresh, sea, and brackish water systems already imagined - although there should be access somehow between the systems which doesn't cause them to equalize, maybe underwater tunnels separating air systems and rooms with air pockets separating water systems, working as crude airlocks. They can even be made on purpose, once a few natural ones let your people figure out how they're keeping they systems separate - and why that's necessary.
And you have a people who move into the upper levels of the cavern system, perhaps to escape some outside threat, or maybe just for better access to the filtered freshwater and the cavern system. Maybe they needed to live underground because they needed all arable surface land (if on an island, for example) to feed people. They might spend generations slowly building themselves deeper into the cavern system - and if they're using freshwater in a more isolated system faster than it seeps in, they might excavate their way downward, using newly drained caves as extra storage and eventually living space, and following the water source to keep moving down. Of course, they can use their technology to make the caverns more habitable, and try to keep the various systems separate but accessible, as they build their city downwards.
The crude bellows and pumps of that age (used for airflow in forges and such) might be not be sufficient to keep the water out of a true underwater city, but they might be enough to keep airflow and oxygen circulating to a livable degree in a deep pressure cavern system with access to the open air at the surface. They understood enough about airflow to to feed the fires with good air in their forges, this is just a bit of a larger scale application to feed good air to their people. Maybe they can supplement their oxygen with cultivated pools of certain types of algae, in places where light might shine for photosynthesis, but perhaps the air is more stagnant - or even using bits of polished metal to reflect light in for just that purpose. They would not be racing race down to build the city hundreds of meters below sea level, they would be slowly, over generations, edging their way downwards. There would be a lot of slow experimentation, and these tricks would let them slowly make marginal areas more habitable, and as they adapted to the area they would find more tricks to make the place habitable, and more tolerance to marginal areas.
Also, as they descend I expect they would be living under more and more pressure - since big pressure differences would cause the cavern system to flood or collapse. But, they might be able to adapt to that too, if the best divers can visit 200-300 meters it shouldn't be impossible for a population to adapt to 100 meters below sea level, just like populations can over time adapt to living at high altitudes. The worst problems seem to some from moving between different conditions quickly, but long term living is easier to adapt - though the best divers from this population might be able to go a bit deeper than from a population adapted to surface pressures. Maybe the population would split into interconnected settlements at different levels, who had problems going too far outside the level they were born in without great care. This might mean after a while it is easier and possibly safer for the divers living at a deeper level to navigate the cavern system to access the ocean (for seaweed, algae, fish, or any other resources) from the deep underwater access point your hero found, rather than ascend through the city and access the water from the surface.
