The machine has to have supported square rails and ball screws to do precisely that job. While you are not aiming at precision the cut must be clear, so the engraving. Otherwise you will waste a lot of time cleaning, filing and polishing .

for xxx $ you can not do that. I would say for less than 3000euro spend, you can not do that, speaking of new machine. It must be small CNC mill or a very high quality small cnc router or you have to build one.

Your best bet will be to find small Chinese mill converted to cnc or another used smal CNC mill. Dont become victim to sellers hype.

Hope that helps.