<< I know that the cotton in the bottle is fairly saturated, but I wouldn't call it "eaten". >>

Probably because it's not cotton, which would be damaged by the free iodine - and consume the iodine. Check again - I bet you'll find that it's fiberglass or MAYBE an iodine-resistent polymer. A little flame on a tuft of it will be revealing - don't inhale any fumes that result.

The straw idea is great. How about some thin wall teflon tubing instead? I suspect that a mechanical seal, such as crimping a metal sleeve, would be required to seal the tubing.

Regards,

Tom