I've tried fatter linen (60/2) and finer linen (90/2). The 90/2 looks closer, and you can see I also tried that with a heavy outline thread (we call that gimp, sorry--I know).
But the pinholes were bothering me. I can get rid of them with some pulling on the lace in all directions. I assume that would be like a washed item might be treated? And why the pinholes aren't apparent in the museum one.
Anyway, more testing to do. But am getting closer, I think.