Hi Nick -
For something like that, either a laser or a CNC can work. I think a CNC machine (like a Taig, for example) is more versatile, since it can control depth better, and you can vary the line-width easier, either by changing tools or by using V-carving techniques and a tapered tool. Your problem might be due to the wood you're using. Something that small would be a good project for endgrain boxwood, which can hold an incredible amount of detail. (If you can't find any, let me know and I'll send you a piece.) If you're doing a lot of this, and want to cut really fine lines, you might consider investing in a high-speed spindle; the single-sided fine-pointed engraving tools that do that work best at very high RPMs.