For a lot of my puzzle pieces, I've laminated 4 x 6mm layers to get the 24mm thickness... though care still needed as while the laser cuts a perfect 6mm hole, Chinese imported dowels are not a perfect 6mm diameter.
Mine cuts 5mm perspex without issues, no melting and re-binding... that's the thickest I have though I use 4mm in my puzzles, with crisp, crystal clear edges, One thing about MDF, when I cut it it binds he glue (or whatever the physical/chemical reaction) and gives a harder edge... which I've found ideal for routing templates. Also if charring is the issue (for solids), I add 0.203mm (my kerf) to dimensions plus an allowance for scraping. Scrapes well, though need to reverse direction when scraping endgrain towards the edge.
If your first cut is too short... Take the second cut from the longer end... LBD
