If you don't mind cutting and finishing, these little rod assortments are quite handy to have around. It's a "hard" brass so it's not the best for some applications (rivets, for example) but it'd be great for this. They're about 4" in length and all four of the kits in brass and stainless I got were accurately sized save for a couple oddballs in the mix. So far all that I've used have looked clean without voids or weird spots inside. Lots of "brand names" for these...I'm sure someone has just 5mm, too, but sometimes just one size or a couple of rods costs more than these assortments.
https://www.amazon.com/Swpeet-Assorted-Diameter-2mm-8mm-Miniature/dp/B07BVRKJZF/