This doesn't really stand alone, unfortunately.

The basics of the story I can follow, and they make for a good one-shot. But it's incredibly hard to make out what any of May's backstory references mean, even though they make up a good portion of the story and, presumably, are relatively important to what you're saying. If you're posting this as a separate one-shot, you should be providing enough information that a reader could understand them without needing to already know the backstory. you're referring to.

"“It’s okay,” she said softly"
Because Quilava hasn't been speaking this entire time, and pokemon speaking abilities vary wildly across fanfiction, and they share the same gender, it would be clearer here if you identified that it is Quilava speaking with some other term than just "she".

The idea Quilava's speaking at all strikes me as really awkward, given that May's entire monologue sounds more like something you'd say to a dog, and if Quilava can make herself understand perfectly, it begs the question of why May doesn't ask her where she'd want to be left or if she'd rather be given away to another trainer. Assuming it's the right place because you see a similar pokemon around and figure they'll be okay with their own kind is perfectly understandable, but for a situation where you can't simply ask them. If it's simply to express that Quilava is okay/accepting this, that's conveyed clearly enough by the rest of that paragraph - she snuggles (acts friendly) then runs off on her own rather than just being left.

I hope this never becomes canon in your story. evil. 8/10 for writing & grammer, but if it did become canon, you might lose a reader.

You’ve sure turned this backstory into a big deal with this little one-shot. My favorite aspect was the portrayal of May’s erratic and confused thoughts. The best example I found was that, even though she /knows/ that Quilava isn’t stupid, she still turns around and punctuates her insult with that word only a few paragraphs later.

The dichotomy of warmth and cold -- excellent usage; you played that theme very well, with regard to both its physical and metaphorical applications. I do wonder though, with Polaryu gone, is Champion Cave going to remain iced-over for much longer? Or is there another reason why it’s so cold?

Lastly, I find myself wondering why this isn’t in the main story. It's distinctly different from "Nightmare" and "Guilty". Is it just TQftL is all about Mark? It certainly stands alone, in any case.

Aw, that's so sad. T_T I had honestly expected a fight or raised voices at least, but Quilava understood May's inner struggle and her charcter quite well. Quilava showed that she's the better being, and actually knew the meaning of forgive and forget. Great job.