Good job getting the LaserAxe machine running. You did exactly as I would have... loading Grbl onto the controller board and then using Lightburn to do your engraving. I misunderstood that you had managed that already... and thought you were still trying to make their restrictive "proprietary" software do something it probably wasn't designed to do.
About cutting materials with a diode laser...
As you've seen, white foamboard doesn't cut with a diode laser... but I should think 1/16" plywood would cut nicely with only a pass or two. I find most often that folks having trouble cutting thin wood/ply/cardboard/etc... are not getting BEST POSSIBLE FOCUS on/in the material. And it's crucial to do so IMHO.
Any time I change materials and/or its thickness, I use a simple focus script to find the Z height that gives best focus on the top of the material. I then set in that Z height and never change it. I then use another script to find the fastest feedrate that completely pierces (or cuts as deeply as possible) the material in a single pass.
This post shows those steps in 3mm cardboard. The focus script generated the "gradient" band where best focus is determined... without a Z-axis this would have to be done manually but it is crucial that it be done. Using that Z-height, the cut-through script generated the ten lines at differing feedrates (1000 mm/min down to 100 mm/min) to find where the earliest complete cut-through occurs. Setting in that focus and feedrate then allowed me to cut the cardboard in one pass, cleanly and without charring.
About needle cutting...
Since your machine doesn't have a Z-axis yet, you might take a look at Edward Chew's
TimSav foam-cutting machine. While he does use 3d printed parts, they are all FLAT pieces... and a DXF or SVG can be fairly easily gotten from them that should allow them to be milled or laser-cut from another material. His machine also doesn't have a true Z-axis and he uses a servo-driven Z-lift mechanism to lift the cutter clear of the material when moving from one part to another... this scheme could also be used with your machine in lieu of a real Z-axis, if desired.
I'll stop here for now. There's tons more to say on these subjects and if there's anything you want me to explain further I'll be happy to help as best I can.
-- David