I have been using 6mm Birch plywood with my laser cutter without problems. Only problem with laser cutting is getting it flat after it warps sitting on the shelf at a store. The flatter the material, the better the laser can stay in focus and cut through the material cleanly. Also look out for knots in the wood - those can be tougher to cut through. I try to pick out my material to avoid them, but sometimes there isn't much choice.
I don't know much about beech wood.
From the specs:
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