The ortur laser master 2 is an inexpensive and effective diode laser that will engrave and etch very well and handle extremely light cutting tasks if you put in some trial and error. You could get a basic one with a lower powered diode for less than $200.
Mine has a """20 watt""" (it's really more like 5.5) laser. I have cut 1/8" acrylic, 1/4 wood, and it can pound out stencils in paper or cardboard quite well as well. Cutting is not the primary focus and should not be the main goal if you consider one of these, but it can be done after a fashion.
It also has a rotary attachment that can be used for round objects. For something the size of a flute I suspect you wouldn't need it though- for things like wooden turned mugs or the like it would be needed.
The last picture is an example of using it to mark an anodized aluminum surface. It is absolutely not powerful enough to cut metal, but you can either cut through coatings like anodizing or paint, or you can use an etchant to get a small amount of depth engraved in metal. I have seen some people use mustard as a medium on steel.
Quite a fun and economical machine. Use the appropriate safety equipment at all times. It would allow you variety in the size of your engraving along with other possibilities, and the lower wattage versions will do just fine and cost less than a branding iron. Mine came with their at the time top tier laser and accessories and was under $400; the most basic model was under $200 last I checked.
