I've cut an interlocking candle holder out of 3mm ply using a jigsaw type interlock (picture here). There is a little movement in the two pieces, but I would expect some movement in a wooden jigsaw; more than a cardboard one. You would presumably be cutting larger pieces than a normal cardboard jigsaw?

As for power; my cheap 40W CO2 laser zips through 3mm Birch plywood on about 30% power or 3mm acrylic at full power.