I thought the Swiss owners had a material supplier thread going, but if they do I can’t find it.
Scrollsaws use the same thickness of wood as we do, so find their suppliers and you should find some 3mm wood.
Proofgrade plywood is wood veneer over a mdf (medium density fibreboard) core. Their veneers are high quality and they then apply a finish and masking. In the US cabinet makers are large consumers of this type of plywood. You probably won’t find it finished, or masked, but you can do that yourself.
You’re closer to the Baltic than we are, so presumably 3 and 6mm Baltic birch plywood is available in France. It should be a pretty cheap and cuts well.
In the US sign makers use a lot of acrylic, so find them. They may have castoffs for cheap or free.
Tiles are a popular material for engraving.
And finally there are people who sell laserable materials on ebay. If you can only find US based ones, many of them don’t know or care about export/import rules so they’ll happily mark a box as samples or something else that’ll have a good chance of not drawing the eye of customs. Of course if it does, I wouldn’t expect a refund.