I'd try the local hardware store somewhere near escudgeon pins, that or just use very small screws instead. small enough the heads wont protrude and interfere with things might need toothpicks glued-dried into the original holes before putting the screws to hold. alternative method might be thin double sided mounting tape instead.

good job making them short enough, that'll work!
local hardware store here has all kinds of small scudgie-pins, isnt too typical either, most might not got.