The story is Raison d'Etre by cywscross.
I had read this story on fanfiction.net, and so all my searches were for that website only. I started going through each fanfiction in the the HP Katekyo Hitman Reborn crossover list on ffn with Harry Potter as a character, and I found one fic where the author had written a Harry X Shamal story, and mentioned that she got the plot from Raison d'Etre by cywscross.
I looked for the mentioned fic and found it on ArchiveofOurOwn (remember - I had only been searching fanfiction.net). A look at the author's page on ffn showed me that they removed all their stories from ffn recently, which is why I couldn't find it. All their stories are now on Ao3 (and tumblr).
Anyway, as I mentioned in the question, Harry comes to be known as QuickSilver in the mafia
"I know what you're doing," He finally said gruffly. "That damn innocent act isn't going to work. Everyone in underground Italy and several other countries have heard of Quicksilver."
Harry blinked blankly at the swordsman, absently tucking his knife away. "'Quicksilver'?"
Squalo scoffed. "That's you, isn't it? Black hair, green eyes, British, Trident Shamal's new partner. Quicksilver."
Harry pinched the bridge of his nose. "'Quicksilver'? That's what people are calling me? Seriously?"
Also, Harry fights with a silver knife, and is very fast, hence the name.
Squalo snorted. "You have a silver knife, trash, and you're rumoured to be fast.
Harry saves Lussuria and Ryohei after their ring battle, ans Squalo gets it on tape. When reviewing, he notes that Harry was not just fast, he was instantaneous.
But Squalo hadn't even seen him move. The rumours all said Quicksilver was fast. This was leagues beyond fast. This was instantaneous.
There was a prequel to this, called "Give It A Year", but it has not been uploaded on ao3 yet.
Thanks a lot Valorum, for giving me that crossover directory link! It wasn't until then that I started reading every story.