He doesn't really look like a "fancy" pirate (unless you want your costume to be more "fancy" looking than the picture you linked to), so I'd be more inclined to go with wool. It shouldn't be any hotter than velvet or another fabric of similar weight. Wool is also the most "realistic" option - though if you really want to go that route make sure you go with a wool suitable for
a coat, not something like your typical gabardine which is rather thin and flimsy.
Since it's not going to be touching your skin itchiness shouldn't be an issue. Most people who are sensitive to wool are actually sensitive to the finish on the fabric, not the fabric itself, anyway. And many wools aren't very "itchy" at all - it's something you should get a swatch for if you're really concerned about it as that's the only way to really tell how it's going to feel. I have a finely woven wool suit that's not itchy at all and the pants are not fully lined and it's fine - my wool tweed jacket is slightly "itchy" if you pet it since there are far more loose, kinky, fibres. However I also have several pairs of wool socks and this wool vest that is very fuzzy - all of which are very soft. So a swatch is the only way to tell, because even a "fuzzy" looking fabric might have soft fibres due to process or finish.
Sorry about the tangent.
Cotton velvet is definitely how you want to go if you want velvet. A synthetic is going to look cheap and fake (they can work for some applications and some are better quality than others but I'm just not feeling it for a coat like this) and silk velvet costs upward of $75/y and a lot of it is VERY shiny and would still not look "right".
You should easily be able to find deep red in cotton velvet or wool and likely the blue too - they're both common colours for this kind of thing and for historical costuming use so they're out there.
If you can't find exactly what you want, you can also dye it.
Actually in the pipes:
- Remaking Ser Loras doublet.
- Luke Skywalker, A New Hope
- Wind Waker starter costume in swim material
Eventually: Paul from Dune, Faramir, Harry Sullivan