Neat project - I’m a little unclear on the problem you’re trying to solve, but I’m assuming it’s that area where the paper came apart.
If so, I think your main problem is that you seem to be way overburning the cut - that’s a lot of smoke coloring around the holes for cutting paper.
My advice is to use the rest of that paper to do a bunch of test cuts at increasing speed and/or decreasing power, until you find the minimum amount needed to reliably cut through the paper.
I would also suggest reducing the size of those circles in the template - if you have a 2mm stone you want to pass through, but your laser produces a 0.25mm focus point, then you’re really burning 2.25mm for each hole - because half the laser is “outside” the area you want cut. Try reducing the size of the hole by 0.1mm and do a bunch of sample cuts on that paper - 1.9, 1.8, 1.7mm…etc. - until you find one where you like the result.