If you are not opposed to going slightly non original as the restoration specialties sample would indicate, McMaster-Carr has some reusable plastic rivets which might do nicely.
mcmaster.com/#standard-rivets/=ksgo2d In the additional rivets section are the removable plastic rivets.
I don't remember the holes exactly but am thinking they are around 1/4" in diameter and the total thickness including brace channel would be around 3/8 to 1/2 inch. The arrow or keyhole types have a couple available in black which might be close in size to what is needed and have a decent looking smooth round head.
If plastic is not to your liking then as you said, it's not something that will come out often.
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