I have had very good results with this sort of material and a diode laser; Little keytags and a couple of larger signs.
I’ve only tried on material with a dark upper layer and white background, start with a test piece and high speeds at full power, then work down to tune the settings. Use a fairly close line spacing, I use 0.1mm lines and 1500mm/min on a pseudo-‘40w’ unit.
The level of detail is good; the biggest issue being scan lines appearing in the lower layer and you should plan on making at least 2 passes at right-angles to each other; with the second faster to clean up the already etched areas… I actually do four; horizontal, vertical, then diagonal in each direction. This is something you should experiment with too; It helps lighten and smooth the second layer.
Cutting out the tags using the laser was possible, but the white plastic is somewhat transparent to blue light and whilst I was successful after many passes, the edges needed considerable finishing. You can see this in the images below, the material is quite durable, a whole summer kicking around in my boat and pocket has not left too much damage. For my latest sign I laser scored out the outline, then cut + folded the material with a craft-knife along that. This produced better results in less time.
The material I use is this: Laser XT Outdoor/Indoor | Duets by Gemini , it’s not too hard to source on-line
Edit: I should also point out that you need good airflow to deal with the fumes (don’t breathe!) but also to stop the flashed off fumes discolouring the cut and uncut material.