First of all, the HORI Real Arcade Pro V3 SA is for the PS3, so you can't count on official support for PC. But it has a regular USB plug and from what I've read online, it works without a hitch on PC, requiring no custom drivers or such.
Secondly, as long as something is recognized as a regular controller by Windows, it'll work on Touhou since it just uses DirectInput (assuming you haven't turned that off in the custom.exe options). I've even played Touhou using a dance mat for laughs. Didn't get far, obviously.
But finally, before you shell out a large sum of money for a really good stick, you need to consider whether it is actually what you need. The HORI RAP V3 SA, along with many other high-end sticks, uses Sanwa parts. These are known for good quality and are also used in arcade machines, but here's the catch... These parts are ideal for fighting games and many arcade style games.
But in Touhou, things are a bit different. The large throw and flexible spring work against you. For those micro dodges and streaming, you'll want something with a short throw, small dead zone and preferably a stiffer spring. It's only from hear-say, but apparently Seimitsu sticks are better for this. This goes for all bullet hell games, but it's especially true for Touhou due to the emphasis on small and quick movements.
So even if you get the best stick around, it doesn't mean it'll improve your Touhou game. And take into account that there'll be a frustrating few weeks of retraining your muscle memory. I've started playing Touhou with keyboard instead of exclusively with controller about a month ago and I'm still not on the same level with that input method.
So here's a couple suggestions for you to mull over:
- Just stick with your current input method. If it's keyboard, maybe look into getting a really good one (fast response, no ghosting...)
- Maybe try something else. Switch to keyboard if you're a controller player, or vice-versa. See how that works for you.
- Get a cheap stick and mod it with parts. It'll take some research, but the separate parts are often quite cheap to get! Plus, you get to choose exactly what you want in there, like the shape of the gate for the stick.
- Find someone with a high-end stick and request to give it a try on Touhou.
Me? I use dirt-cheap controllers that cost less than five bucks a piece. They only last about half a year on intensive play (d-pad tends to deteriorate) but they're just the perfect thing for Touhou. I buy them by the heap. Never found something I like better for my games.
So, don't expect that high quality or a high price is going to reflect upon your performance. At least not immediately; maybe over time and if you're a real high-tier player. Most of the time, though, it'll take a lot before the hardware is what's keeping you back from improving. However, if you're planning to play other things as well, or you're also into fighters (and hey, with Touhou 13.5 announced today, now's as good a time as any), you'll probably get a lot of bang for your buck. And high-end sticks are also really durable, so it's an investment.