Yes, I'd try the Dawn to give her a good cleaning up.Not totally certain if the long piece is tissue. Is there some sort of test I could do or other way to determine it?
Also, would it be bad to use Dawn dish soap on the tissue? It works like a charm to get gunk off of chickens, but I have concerns over using it on internal tissue as I've only ever used it on their scales and sometimes vent feathers.
Just reapply your ointments when your done.