Practice using about 150 grit sandpaper that is glued to a flat board. You should be able to the get a fairly even taper to nothing. If you use one of the new cyanoacrylate glues that are colourless and set up in a few minutes and elastic tape like electrical tape as the clamp, it might work.
If you need the wetting of a water based glue to make the veneer pliable, pick one whose colour blends. Maybe a few wraps of veneer.?
I did some perfume bottles a few years back and had problems with late failures from wood movement around a fixed tube. Good luck.