ok, so for reference it looks like music wire is high carbon steel, whereas medical wire is either stainless (sometimes braided stainless -- OUCH), or a variety of precious extremely hard alloys that mostly involve titanium. OUCH.
high carbon steel is not a huge deal, but you are only going to be cutting that with a carbide cutter and if you want it flush, you're not going to get a whole lot of cuts out of any tool before you need to sharpen it. stainless is trickier. carbide should work but again, fewer cycles before you will need to sharpen depending on the alloy. hence, you want full carbide cutters, not carbide coated (in fact most of the precious metal tools out there, nit/tin drill bits for example, are only like 9-10 microns of coating). yah.
i would say the best way to go is a big fat gigantor cutter or anything that will make the cut, then abrasives to get it flush. a non-flush but effective cutter and then working up to very high grit abrasives is going to get you an even better flush finish than anything but the sharpest highest leverage pure alumina forged from heaven flush cutter on the planet.