So he doesn't actually try to do that. The piles of rats are a distraction, an environmental hazard that he needs to work through and past to get to his actual foe. He'll slice, of course, kill them in droves as he moves, but every movement has a goal: killing the tiny vermin. And surviving to do it, of course, his own safety is paramount here if he wants to leave, but assuming he's surviving he is aiming for the tiny vermin.

He kills the tiny vermin. Zagreus, god of he's not going to learn it's actually life and blood and rebirth until much later, which means he is quite literally literally god of leaving the Underworld, exterminates the small, unreasonably powerful rodent. And his intuition was right, for as soon as he succeeds, the hordes of other rats stop in their tracks as if suddenly free of some form of external influence and, upon noticing the myriad bodies of their comrades littering the floor conclude they should flee if they wish to see another [day? night? unit of time].

There are still some rats that try to take lucky shots at Zagreus, but none of them get close to him, and soon enough the chamber has only him and the corpses.

And he wants only one corpse, the one corpse that will serve as his trophy to prove to Cerberus he's helped with the problem of vermin. ...and maybe a sack filled with bodies of other vermin just to make sure he has enough proof.

The way back to the entrance hall is relatively painless and short.

K̵̡̡̨͈̞̗̥̳̞̫̥̲̩̥̥̀́͊̀̂̉̉̏̒̉̈́̚h̷̢̬͙̰̲͚͈̘̱̲̓̽̈́̃̓̌̈́̀̀͆͗̾h̶̨̢̻͔̦̩̪̩̼̗̫̬͕̙̉̆̌̈́͌̃͛͐͂͗͝ḩ̶̛̛͔͓̃͋h̴̡̨̺̞͎̬͚͗͗̈́̒̈́̌͘ḣ̴̨̨̧̛̟͗͋̄͌̄͛̈́ͅr̴̛͚̙̪̝͒̆̍̾̆̃̈́͠ę̵͚̭͎̮̩̞̪̟̫̐̏̔̄å̶̛̭̯͚̇̅̂̊͋̍̎̈́͋͋̀͘͝h̶͍͈̗͓̱͔̊́͆̀̋͌̀́̚̚͝͝h̸̞̥̣̯͊̍̈́͒͆̾̈́͘͠.̴̢͙̳̹̰̻̻̩̰̣͙̼̇̓͌͛͗͐̔̈́̓͝͝

"Why, thank you, Charon, it's good to have your support," Zagreus replies, (correctly) assuming Charon is wishing him good luck as he walks past the boatman towards the three-headed dog.

The three-headed dog is delighted to see him again and expresses this delight in his usual slobbery fashion. But most importantly from its reaction to Zagreus's offering it definitely recognises the tiny vermin. Seems he's been having trouble with that rat for a while.

"Happy to be of help, boy. Going through all of that, honestly I understand why you hate these pests. Hope this lightens the load."

Bork bork!

"Now... am I allowed through?" he asks, trying for nonchalant and failing miserably to conceal his eagerness.

Cerberus barks again but then sniffs suspiciously at Zagreus.

"What—oh. No, Yvette isn't here this time. I... don't really understand why, but she has her reasons I'm sure. You'll probably run into her on your way back to the House."

Bork!

"Yeah. Anyway, uh..." Then he stops and looks beyond Cerberus, at the enormous double doors that are the only thing that now stands between him and his destination. "Here goes nothing."

Cerberus whines but lets Zagreus past, watching him intently as he pushes the doors open and steps into the cold, snowy outside.

Zagreus steps forward, his flaming feet leaving melted footprints in the snow. He looks up at the sky—the sky! He's heard of it before, from Achilles and other shades, the endless expanse above. It's... darker than he'd expected, really, not the clear blue he'd heard daytime would be like, so perhaps it is night.

The doors open to a small clearing, surrounded by boulders and ruined pillars, with a clear view of the frozen surface of the river that becomes the Styx.

But between Zagreus and that view...

"Oh. Of course."

Hades has his back to him and to the doors, seemingly watching the horizon. "...I never grew accustomed to the air, up here. It gusts senselessly whichever way it pleases. I suppose you must prefer it to the stillness of the air below."

"You know, you're right. The air up here seems pretty good so far. But I will reserve judgment until I've had my fill. I have to say, though, Father: the Helm of Darkness suits you." Another relic from the war with the Titans, just like his weapons. Zagreus didn't think his father would touch one, given how much Hades seems to hate the Titans, but he supposes the circumstances are unusual. "I'm touched you'd dust it off on my account."

"It should not have had to come to this. For all your antics with me and around the House, I always kept my temper, have I not? Unlike you."

"My temper—!" He takes a deep breath, then resumes in a more still tone of voice. "Well, let's see. You've berated me repeatedly and often. You've lied to me. Mustered a bunch of incompetent wretches to try to kill me. But, sure, I'll grant you that you've always kept most of your anger bottled up. All that's about to change, then?"

Hades takes a second to reply, and inhales deeply before he does. "We're gods, boy. Killing one another is our lot. Although I always thought I was above it. I told you that you cannot leave this place. It seems I must enforce my rules myself. I'm sending you home, now."

He turns around, then, and shrugs off his cape, which dissolves into fire. His trident appears out of thin air, and he grabs it, assuming a ready stance.

"...no. You're going home, not me."

His shield is ready. As ready as it's going to be.

It will not be an easy fight.