There are a few posts regarding laser cutting of plywood and searching through the forum posts should be helpful. In my recall; the 1.6mW laser is able to cut through 2mm thick plywood. Here is one post that mentions settings: Laser cutting plywood
In theory, if the laser is strong enough to cut through a single layer of card, then a second and subsequent layers should not present any difficulties to the laser beam. Feed speed can prevent edge burn and smoke damage as can power. It is helpful to use full power and to move the laser module head fast enough to cut and not so slow as to cause burn damage to the item being cut.
This is likely to need a lot of experimentation with dwell time, power and feed rate. Luban does create a test pattern of boxes at different powers. Do ensure that your settings are correct in Luban. You specifically need to check the multiple passes box and should have entered a value in the depth of passes earlier in the setting dialogue. I mention this because it is easy to miss the setting for multiple passes. I believe it is not checked automatically.