I need a slightly sturdier, thicker piece of wood to help fill in my inset. If you have a similar box, follow along...
Use your cut piece of embroidered balsa and trace it onto your thicker wood to get exact sizing. Now, cutting it involves a little trick. I’m a crafty person to be sure, but on a day to day basis, I find myself not in need of carpenter tools, so I didn’t have a wood saw about. What I did have, was an exacto knife.
To cut a thick piece of wood, score the line with your knife, and then go over that line again and again until you’re about halfway through the wood. Then, using the marks on the side, score a mark on the back of the wood that matches your line, and snap it in half. Ta da! You might need to sand the edges a bit, but it should be a clean enough break.