I'm not against the reamer idea, but if you do want to make the spindle smaller, I'd go with sandpaper backed up by a flat surface (like a file) as suggested above. I have quite a selection of sandpaper, so I would first start with 1000 grit (wet) and sand it for 15 seconds and then re-measure. Once I saw how much I removed, I would either continue as it was removing a good amount or I would drop to 600 or 400 grit (wet). I would switch back to the finer grits as I got to final size, because part of analog audio is beauty and sanding marks on the spindle is not acceptable to me. I have 2000 grit and 'mirror fine' paper as well, but understand many people don't have this kind of selection.
If you do decide to go that route and have a local mom & pop bodyshop that does nice paint work, they likely have tons of abrasives in the right denominations and if you asked nicely for a handful of pieces they might sell them to you vs having to by 10 sheets of each. Remember I said I had tons of sandpaper. . .its just because of that. I was doing my own motorcycle helmet paint job with automotive paint and clearcoat and bought all of the grits I could from a paint supply store. I think I had $50-75 in sandpaper alone, since each box was at least 10 sheets. I can't imagine what it would cost today. . .
But, the reamer is a valid solution for sure. I'm curious what the ideal flute configuration for thin vinyl might be. Definitely a straight reamer, but I wonder how many flutes. I've never reamed plastic, just steel, aluminum and bronze and each one of those definitely liked different types of flute profiles.
Sandy.