The first 'round' of my campaign will end with the deaths (and resurrections) of all of the players. Their role before catastrophe strikes is as teachers in a philanthropically funded school in an impoverished, war-torn province in a crumbling kingdom. There will be at least one dungeon in this round, and the players will have an initial quest that involves clearing it (with the real stakes of the quest only becoming apparent once they've started it)

I'd like to know what is a good way of thinking through challenge and enemy design such that the enemies are in a sweet-spot of difficulty: not impossibly hard to challenge that a TPK would result, but also clearly hard enough that even when the players 'win', they slowly realize there's no way they can make it through the whole dungeon?

Additionally I plan on reconfiguring the players' starting stats after the resurrection. If there is a 'sweet spot' of starting level vs enemy level that works (like lvl 3 vs level 4 is a better uphill struggle than lvl 1 vs lvl 2 or 3), also include that in your answer.

You don't have to use everything that you prepare

It's been my experience that it is much easier to leave something out than it is to make something up on the spot, so my advice is to, for lack of a better word, "over-prepare" in this particular scenario.

That is, build your encounter as if you're planning a TPK, keep that material on hand, and then leave most of it out at first! If your party starts to just handle things, bring in some of that extra material, and keep doing so until you've figured out the right balance between "challenging" and "impossible".

As to where that balance is, I can't actually offer much advice. That depends on too many factors, such as party size, party composition, players' mentality and tactical capabilities, your own tactical capabilities, and so on. So, you will have to experiment with that on your own.

This approach has the benefit that you won't find yourself underprepared, meaning you can always ramp up the challenge level! And conversely, if your party is less tactical (or more unlucky) than anticipated, then you're still reasonably covered from a TPK!

But how do you make it clear in game that they're not supposed to win?

If the players are practiced, they should be able to size up their opposition from that first encounter and decide "This is a bad idea." But if they're new, or if they're stubborn, then you can outright tell the players that their characters are fighting something out of their league, and that they will (almost) certainly all die if they continue. It's a pretty sensible conclusion that these adventurers would be able to recognize when something is just too much for them, and it also gives players the right information to make an informed decision.

A word of caution

Looking from a player's perspective, this feels like some heavy-handed railroading (that is, the players might feel like their choices and actions in this session don't matter). This can leave a poor impression on your players if they aren't expecting it, and many players don't like to have their agency taken away.

I would suggest letting your players know before-hand that this session will end in a TPK, but that they'll still be able to keep playing these characters. It might ruin the surprise, but player buy-in often helps things go more smoothly as well, in addition to building trust between player and DM. It's been my experience that players are willing to accept less agency when they know that it's part of a story's setup.

Exceed their adventuring day capacity

To achieve the effect your describing, starving you players of resources are likely going be your best bet. First level characters can handle 300 Adjusted XP (per character; DMG p. 84) in a single adventuring day. This also assumes a two short rests, and so denying them those will wear them out faster. Once players are running low on resources they will likely try (quite understandably) to take a Rest which you will need a plan for.

A hard encounter will usually (subject to dice rolls) tax some healing resources (i.e. spell slots or hit dice), so having your players first have an encounter that tells them the respective creatures are dangerous, and then show them you should be able to invite the feeling of hopelessness you described.

However (speaking from personal experience) deliberately killing your party, while it can setup certain stories, need to be done with a lot of care and I would absolutely give some premonition of the pending resurrection before they even begin the dungeon.

What you're aiming to do is have the players (and their characters) know that they're not going to make it to the end alive. This means they need to learn 2 things:

1. How long the dungeon is. How can you know you're not going to make it to the end if you don't know how far away that end is?

I would share this information in a way that the NPC's say that the old manuscripts state that it used to take X days to travel through before it was made too dangerous to use. This is essential information if players are to know they can't make it to the end, as they may think after every fight that they're nearly there. You want to highlight they're really not.

2. That the difficulty is getting harder - and will be impossible to win by the end.

There are 2 ways to do this (which can be mixed) that I can think of:

a) start easy and having a notable gradient. This gradient involve must be a TPK by the time they get to the end; and a viable first few fights so they can establish that it's getting harder.

b) Prevent the players from recovering from an encounter. Have small parties attack them at night so they never have a good nights sleep. Mages will run out of spells, fighters will start to run low on hit points. Fatigue for everyone. How long players can last this will vastly depend on their level and equipment - so they may learn very quickly they're not going to get to the end; or it may draw out if they have potions.

If you're going Demon's Souls, go full Demon's Souls.

There is nothing you can throw at your party to kill them if they roll all 20s and you roll all 1s.

There is no guarantee that won't happen, because math is cruel and D&D doesn't care.

So take inspiration from the game that sparked the "you die and then" craze, which threw you into a fight in tight quarters with a demon that could take a lot of punishment and kill you in three hits. You could beat it anyway, if you were lucky or good, and loot its body and the room beyond, but then you'd have to open the door to a dragon's killbox and die in a cutscene.

Make the dungeon hard! That might kill them, it might not. But after the hard part, and after the rewards they get from surviving it, put something in it that will kill the PCs. Not even play-it-out roll-saves take-damage kill, just kill as a story effect. They have to retrieve the Orb of Zot? Shanksworth carefully lifts it off its pedestal, there's a sudden pulse of darkness, and then the PCs wake up wherever PCs wake up when they die.

Fudge Rolls and Stats

The easiest way I can think of is kicking dice rolls for the monsters as high as possible and bringing up stats, like AC and attack damage by a few points.

Higher CR

Most of the time if the encounter is supposed to be extremely challenging but is akin to a single-boss fight, I usually use a single creature with a CR around 1 CR higher than my party’s average level. (2 CR is pushing it a little to deadly, but may still work if you tip some dice rolls to the party’s favour.)

Reinforcements

If the encounter is a fight against a large number of enemies, then make waves of reinforcements. If the party can finish one wave, throw another at them with more enemies (ramp up the number gradually, not too fast) until they realise that they can’t keep up with the endless flood.

Waves and stages are my take.