The laser cutter cuts in 2D, so you wouldn’t be able to do that particular style of dovetail, but you could do a finger joint which is quite similar:
(This is the model of a toilet stool I made using 1/4" birch and finger joints on the laser.)
Also, baltic birch is typically available in (nominally) 1/8" (~3 mm) and 1/4" (~6 mm) thicknesses. Note that the finger joints will fit best if you base the design (i.e. the length of the tabs) on the actual measured thickness of your material.
Edit: Forgot to finish note on etching. Definitely something you can do on the laser
To use the laser, you need to be a member and do the training ($30, ~3 hours, typically offered every 3-4 weeks).(Add your name to this thread to express interest in training: Anyone want laser cutter training?) You may also be able to connect with a trained member to do some cuts on your behalf while you’re waiting for training.