Hey Nonnie!
Well, in canon it’s not presumed to be anything, really. As you say, it’s most likely a working relationship.
In Sansukh, they are very close. But it isn’t known what the nature of their relationship was. I am not confirming whether it was platonic, romantic, or whatever.
This is quite deliberate, and I am doing it because this is how Narvi is (or at least, my version of Narvi). It’s a conscious character choice. She is something of an enigma, and she does NOT talk. About things. That hurt her. She holds things close and tight. She keeps her secrets.
Check out this answer in particular, and also my Narvi tag for more!
This is quite a point of difference between her and all these other focus Dwarves, whose personal lives end up splashed all over the walls. Narvi could not bear such a thing. GOD no.
The closest she has ever been in the whole story to letting out anything personal is in Chapters 32 and 36… and each time it’s only one line.
Chapter 32
“I’ve seen good, innocent, true-hearted people hardened by the world,” Narvi said, and her eyes were dark and glittered strangely as she watched Pippin. “I’ve seen naivety punished and gentleness broken. It’s not a new tale.”
..
Chapter 36
Gróin and Fundin were, for once, sad-faced and silent: not a single bickering line passed their lips. Haban was tight-lipped. Narvi’s eyes were full of ancient rage and hatred, her hands clenched by her sides.
“There is nothing I would not do,” she began in a voice that shook with fury. Then she checked herself and stalked away.
“What-” Fundin said, scratching the back of his head, but Haban held up a swift hand.
“There’s not a single one of us,” she said quietly, “who can even come close to hating Sauron as much as Narvi does.”
…
So yeah, the relationship is never going to be defined. I am fairly sure that for Narvi, it would be indefinable anyway.