You’re asking about kerf reduction. (Popular topic.)
If you want to fit something without the gap caused by the kerf, you will generally have to adjust one or the other of the pieces so that they fit snugly. The laser beam burns away half of the kerf on either side, so the plug piece that you cut out of a hole is going to be a little small, and the hole is going to be a little large. The way around it is to cut the plug for the hole in a different area, make it just a little larger than the one designed, and it will drop into the slightly larger hole and fill it up completely.
You can adjust in either Inkscape or Illustrator, depending on which you are more comfortable with. There are discussions on how it’s done below:
The kerf for veneer is going to be fairly straight, so a simple adjustment should do just what you need.