It's on the tip of my tongue... It ran for many chapters (I think, to a reasonable conclusion), as well as a sequel, and a side story in "the other pants leg of time" (yes, they acknowledged borrowing from Discworld). The basic premise of it, as I remember, is that yet another bizarre martial artist shows up, but in a twist, he's after Ryoga. He also wields an umbrella, but his is a fancy black one, which he wears with a dapper suit (at one point, he points out his annoyance with how his theme results in him having to replace a lot of suits due to fights). He has a younger sister, who has a bit of a crush on Ryoga. When he explains why he's opposed to Ryoga, I seem to recall it more or less came down to that Ryoga showed up at the school (entitled something like "The School Of Unorthodox Combat Weaponry") and learned the umbrella techniques effortlessly (umbrellas were only one branch of it, although I remember them referencing The Penguin and I think Shizumaru Hisame as people who trained, or had trained, at the school. His little sister did some other weapon).
After much fighting back and forth, with the new antagonist doing his best to maim Ryoga while Ryoga treats it as friendly combat just like it always is in the series. Bits that I remember include an all-night Jackie Chan film festival with most of the characters attending at one time or another between fights (Ukyo dressed up in a trenchcoat for it, I think to watch Ranma and Akane on a date?) and a climactic battle on a bridge between the new guy and Ryoga.
That battle was a big set-piece in the fic. They meet on the bridge and respectively destroy opposite ends of the bridge using the Exploding Point Technique (the rival had picked up the Exploding Point technique by watching it being done, and had thus missed out on the toughening aspect. He was very disappointed to find out it didn't work on flesh after he tried to poke Ryoga with it). During the battle, there's a part where the rival throws his umbrella much like Ryoga does, and it's noted that his technique isn't as good, which means the umbrella kind of loops around unpredictably. And near the end, they wind up doing multiple chained suplexes to each other before the bridge gives way and they're washed away.
For a while, both of them are gone with the younger sister wearing Ryoga's bandanna in memory of him. There are interludes involving amnesiac umbrella-wielders where it's left ambiguous as to which one of them it is, including something where they run into M. Bison and accidentally get some of his Psycho Power. At some point, they both come back. The rival has been working as a criminal enforcer (I think as some sort of hitman) and is now able to concentrate that Psycho Power into the leather gloves on his fists (there was a scene where he stood on a rooftop and continuously "flexed" the power into his fists to get stronger) and has also figured out a way to use the Breaking Point Technique on living things (he has a needle point on his umbrella, and triggers the Breaking Point on the bone).
The spin-off fic (or maybe it was the sequel) had the same characters, but were traveling through dimensions due to a locket of some sort.
I read this probably in the early 2000s? It was on an author's website, but so many of those have gone down over the years. I don't remember what other stories might have been on the site.
Another thing I just now remembered, the author also did an MST of the infamous fanfic "Ukyou Can Cook" using the original characters from this story.