Stitching sounds very sensible!
Lace is made from cotton; you could probably find the exact same cotton if you wanted to, but the same colour would be fine as long as it's not too much thicker. You wouldn't need to pierce the thing, your stitches would go over strands in the lace.
Piercing mount board with a sewing needle can be a pain, literally! I once pushed a needle through my thumb - hit the back of my thumbnail, and that was using a piece of leather to push with too, so it went through that first!
After that I used a thimble, a thumble, a finger guard or a gripper, which is just a disc of thin rubber that enables you to grip the side of the needle; not the end. It's all very hard on the wrists though and you'll bend plenty of needles.
I also fashioned myself an awl, snipped a needle to about 20mm, drilled a tiny hole in the end of a short piece of dowelling and stuck the needle in it with epoxy resin leaving just 5mm or so poking out - great tool.
These days though, if my micro tag gun or normal tag gun doesn't suit (either makes a good 'awl' for normal stitches too BTW) I'll probably use Melinex straps that can be cut so narrow that if you dropped one you'd probably never find it again!
With stitches you ideally need a third hand to tie off behind, and normally two holes in the mount board, one in; one out, with the 'in' being made from the back of the board. On this type of thing both ends of a tiny strap can go through the same hole (slit, better) and splay out each side of it behind. No bulk (knots)either.
There's no 'arm in either - that was off the cuff.
