Fate is a great system choice when the narrative is more important than realism. If this is true of your group and the story you want to tell, then you can use it to great effect for your dungeon crawl, otherwise you may want to shelve it for another time. That said, a narrative-driven game is necessarily telling a different story than a numbers/realism-based game is, and you will focus on different things than "did my broadsword attack exceed his defense?" Your job will be to complicate the characters' lives with plot, more than with armies of bad guys (though sometimes the two collide in a Venn diagram of plotty battle).
there will be very little human interaction until the end game, and then it will be with people who necessarily have no background connection with the PCs.
@BESW and I ran a short Fate Accelerated session in chat the other day in which there was no human interaction at all. The session consisted of narrating the character's journey as challenges and contests, with quite a bit of discussion along the way -- one of Fate's defining features, in my experience, is the amount of narrative control that the game gives to GM and players alike, which will be something to consider for your dungeon crawl as well: your players can help you to give them the types of encounters that they're interested in seeing, but it does change the social mechanics at the table, especially for groups that are primarily interested in "winning and loot" or have become accustomed to the GM having absolute narrative control outside of player actions.
A quote from the Fate Core chapter "Scenes, Sessions, and Scenarios":
The best scenarios don’t have one particular "right" ending. Maybe the PCs don’t resolve the problem, or resolve it in such a way that it has bad repercussions. Maybe they succeed with flying colors. Maybe they circumvent the problem, or change the situation in order to minimize the impact of the problem. You won’t know until you play. (FC 226)
To emphasize, one thing that helps with running an interesting, narrative-driven session is to give up the idea of planning every eventuality in advance. Or even some significant portion of them. Paint some broad strokes, and be prepared to work with your players to tell the story, instead of trying to corral them into the adventure you've planned. That transforms a dungeon crawl from a series of rooms with monsters and treasure and traps, and into a living thing that adjusts and adapts to the story that's unfolding. Which is, again, not to say that the story can't unfold into a bunch of angry orcs looking for blood, but the focus should be on the dramatic tension of the situation, not the logistics of the impending combat.
The challenges they face will be typical dungeon challenges - traps, combat, and navigation - and very little in the way of group cohesion challenging interaction.
This can be just fine. If your source of drama and interesting narrative isn't coming from tension between player characters, make sure it comes from somewhere else. Fate characters are all highly competent and motivated, but flawed or otherwise vulnerable to complications in their lives (in the form of their trouble and compels on their aspects) as well. Use this to your advantage. "Because you're Easily Distracted, it makes sense that while you're keeping watch in the night, you become enthralled with some rare fungus growing on the cavern wall, and don't notice the ratmen sneaking into camp to steal your supplies. Damn your luck."
Another tenet that I've gleaned from the Fate books: only roll when failure is interesting. If failure to accomplish some goal stalls out the story, it should probably be glossed over or omitted, or better yet, changed so that it doesn't stall the story. A failed navigation check that ends with the party hopelessly lost and unable to continue isn't very interesting. A failed navigation check that ends with the party at the business end of a lot of pointy sticks wielded by angry denizens wondering why intruders have stomped all over their sacred burial ground is interesting, and can lead to lots more great narrative (or at very least, lots more flashy combat).
A lot of dungeon crawls (in my experience) come replete with binary pass/fail conditions: I failed my dexterity check, I got hit by the spear trap, I take damage. When telling your dungeon crawl story with Fate, be sure to think in terms of "how does this encounter make the story more interesting?" If it doesn't, change it so that it does, or replace it with something else.
Coming from a D&D background, I always wanted to tell more cinematic, narrative-driven stories, but was constantly frustrated by my attempts to do so within D&D's mechanics. With Fate, I've had to let go of a lot of concepts like collecting loot as a reward (unless the loot itself is interesting to the story. Think Sting or Vera or Glamdring or the Elder Wand, for example), but I've found great success in telling the kind of stories that I want to tell. By adjusting the direction from which you approach your dungeon crawl, I think you can find similar success as well.