Yes, it is possible
Jered Adams shared, in a Youtube Video from 2023, that he could, in fact, engrave prints with a laser, and his method. The problems he encountered were mostly related to getting good reference points and affixing the part. He did use a large industrial-size machine with a 40 W CO2 laser module. For the positioning, he used a printed block of known height to adjust the laser module. After making an initial high-power pass with air assist, he added powder-coat paints to the depth to make the engraving easier to see and turned off the air.
During first engraving with 12 %, notice the thin white smoke at the center where the laser impacts, getting pulled away by the air assist. The engraving itself is visible as a slightly different color.
After adding the powder and burning it in with a lower power setting (9 %), the result is very legible:
Power settings
Mr. Adams used 12 % power on the initial pass and 9 % on the "inking" pass. Taking his 40 W Diode as a base, that'd correspond to 4.8 W and 3.6 W respectively, so could be done with a 5 W laser module.
Do note that the initial pass power could also depend on the color of the laser and the exact composition of your polymer. This is because some colors and materials absorb wavelengths from the laser differently, so it is hard to say for sure the same setting will work for all colors the same.
Print settings
The engraving, with the right settings, should at most be one wall thickness deep, which would compromise that one wall's carrying capacity. A 2-wall print might see significant weakness due to the engraving, but with 3 or more walls, the impact would be massively reduced.