How important are your blacks? I find matt papers are much gentler with them than anything with a shine.
Never printed on metal so re: paper....
Ultimately it's difficult because it's
your interpretation of how you want the image to look - especially the reds and oranges in this image. Generally speaking (and to my eyes!), a matt paper will mute (i.e. soften rather than desat) the colours and edges which will look great for the neck and softer feathers but will mute the lovely sharp edges on the wing plumage.
Shiny papers will define the edges really nicely, but possibly over-define the softer areas. Shiny papers also have that gloss coating which reflects the light causing "shine" from different viewing angles. I include nice cotton baryta's in this too like Canson's Platine Fibre Rag.
How big are you going? Is the image going behind glass?
With all the greys in this I'd try Hahnemuhle's Photo Rag Metallic. It's a gorgeous paper.
Do I recall correctly you're in the US? If you are,
Museo Silver Rag is another wonderful paper, but we can't get it in the UK any more