Following the answers and some experiments, I coded my own halftone filter in OpenCV inspired by this
method. I put the code on Github.
For the particular photo of the bunny, with 3x3 tones and the blue channel,
I got a half-toned image where, although the bunny is noticeable, it has poor
contrast:

I was unable to find good settings for the particular wood and laser cutter at
hand. Below is a photo with power 22, 40, 30, and 35 (the photo is rotated 90º clockwise, but the engraving is such a big lump of char that it doesn't matter):

Note that the result depends on the power of the laser: if too powerful, the
engraved black pixels will burn the nearby white pixels.
I then tried several settings with a photo of a show where I did some engravings for the designer (copied with permission from the author):

I tried half tones of size 3x3, 2x2, and 1x1 (which is the same as a hard
threshold), the standard conversion to grayscale or only the red channel (which
worked better for this particular photo), and whether the format of the image
was 8-bit (with 256 colors, even though I only used two, 0 and 255) or 1-bit
(with 2 colors). I also tried the Photoshop half-tone filter suggested by user287001 from here.
I used pine wood for these engravings, unlike birch ply for the bunny. From left to right and top to bottom:
- 3x3 tones, standard grayscale, 8-bit image processed by LaserCut Pro
- 3x3 tones, red channel, 8-bit image processed by LaserCut Pro
- 3x3 tones, standard grayscale, adjusting the brightness spectrum to have the same amount of each tone, 8-bit image processed by LaserCut Pro
- 2x2 tones, red channel, 8-bit image processed by LaserCut Pro
- 1x1 tones, red channel, 8-bit image processed by LaserCut Pro
- Photoshop half-tone screen 72 ppi, line, 5 lines per inch, 45 degrees, 1-bit image not processed by LaserCut Pro
- 3x3 tones, red channel, 1-bit image not processed by LaserCut Pro

For pine wood, I got the best results with a threshold (1x1 tone) which gave crisp edges such as fonts; and with the ten 3x3 tones, which gave good shades of grey for photos with. I felt happy with the last result (3x3 tones with red channel) so I stopped the experiments and published the code.
I also tried on the computer the Photoshop engraved effect action from
here
and did not engrave it because it was a grayscale effect. I found fake engraving in
GIMP
and it seemed similar to Photoshop's half-tone screen with lines, so I did not
try it.