Hmmm...
You could possibly use a stone grinding bit (y'know, the ones that look like lollipops
) to recess the top of the hole for the 'countersink'.
Hole punch and a hammer, maybe? May take a good coupla whacks to get through the plastic core tho.
Either a proper hole punch (like for fixing eyelets and press studs / rivets, etc) or just make one with a short steel tube the right size (drop in a nail at the top to hit with hammer). Maybe sharpen up the edge with a file first.
Could try fabric rivets to fill the holes up - like the ones holding yer jeans together.
Or those plastic rivet / plugs that hold some goggles' foam parts together? Like on old-schools.
Just a thought.
C.