Unfortunately, the success or failure of a deep engrave is going to be somewhat material dependent - that stuff looks pretty rough. If the glue is inadequate, you might get some flaking on it after the fact.
I see that you’re running tests on it which is great, it’s the only way to lock down the settings that work for any particular material.
What I’d suggest is to keep running engrave tests until you find some settings that do not cut through the veneer. It’s going to look more uniform that way.
The slower speed settings are going to cause charring, and that also causes flaking. What you’ll want to use is 1000 speed for your engraves, and start reducing the power to engrave less deeply. LPI should probably be in the 195-270 range to keep from cutting too deeply.
You can also try full power with a defocused engrave - it will cut much more shallowly. (Add 0.1 inch to whatever the actual thickness of the material is.)