Over the years, I've not only gotten my OWN HAIR stuck in fly paper a bazillion times (you'd think I'd learn), but years ago, I had a baby cockatiel that took its first flight, smack dab into a strip. And of course, he thrashed around and got tangled up in it within the 1/2 second it took me to run and grab him... He became "Sticky Bird" and was with me for 13 years.
With my hair, I just wash it several times, and have had to trim out the little bit that was left a few times. With Sticky, nothing helped - he got less sticky with time, but he looked like hell til he molted.
I think I would try washing the little one, maybe with Dawn dish washing liquid (its used for wildlife caught in oil spills), and maybe keep it separate for a few days.
Good luck!