I ran a job kinda like this for a medical research lab. 1/2" delrin,
3"x4" plate, 2mm holes on 4mm centers, in a grid pattern. The kicker:
no burr AND no countersink permitted on the hole, top or bottom. The
grid is being used in a MRI machine, and they fill the holes with an
MRI-opaque gel, which projects a grid of dots onto the MRI image.
They fill the holes by spreading on the gel and then scraping it off.
Since its a shadow that's being projected, the size of the dot is the
widest part of the hole, thus no countersink in order to maintain the
hole size. I drilled the holes on my Taig CNC, with a 135 degree
split point drill bit (brand new, screw machine length) at 5 IPM, 7500
RPM, no peck. Still had a few minor burrs. The part got wet-sanded
on the surface plate with 600 grit SiC paper. This worked extremely
well for the one-off, but not practical for production.

One idea: modify the drilling fixture to drill through another piece,
preferably of the same material. Similar to using an extremely close
tolerance drill bushing. This will hold the burr down and the machine
won't see a change between the two pieces.