The packaging tape I use on the ud or cumbus is a lot springier than polyethylene (it's reinforced with fibreglass or kevlar). It comes in several different strengths (for bundles of plastic pipe, timber, concrete slabs, whatever) - look in scrap bins at building sites. The harder varieties take a while to shape and smooth (use an emery board), and you need a polished surface to eliminate friction noise, but they give you a lot of options.

I don't think you could disguise any of them as feathers, though.

Incidentally, the usual Turkish grip on the ud is different from the Western lute pick grip. To get started: hold your hand palm-up, lay the pick across it along the crease at the base of your fingers, on top of your thumb as well, so it hangs out both sides of your hand (or at least at the thumb side, if it's a bit short). Curl your fingers over it, bring your thumb up to flex the tip upwards, then turn your hand over. At the end nearest the instrument, the pick is fixed between the distal joints of your thumb and forefinger, *not* the tips, and the rest of the pick in anchored inside your curled fingers. Adjust the grip so that the pick hits the string with its flat side parallel with the string and its axis at 45 degrees to the string. At first sight this seems hamfisted, but it's far more relaxed than squeezing with finger and thumb tips and gives you just as much control. I think this should be explained in one of Cinucen Tanrikorur's YouTube tutorials.