Okay, you’ve already figured out that turning of Shadows (in both feathers and bird) helps.
I’ve reduced it to the basics and gave the feathers a Transparent Shader and you were absolutely right @bkol that the lines are caused by the intersections.
With Shadows off:

compared to with Shadows on:
Increasing the resolution and the filter size also makes it a little better, but seeing that this is for an animation, it’s not worth the extra render times:

I’d actually sooner opt for a lower resolution, which also improves it a bit (maybe in combination with a denoiser ?)
Anyway, I’ve been trying things out and I didn’t find a way to completely remove those artefacts, sorry.