Hi Lerameur, I own a machine shop, and modify enclosures for customers from time to time. "Bud boxes" are great, I have no idea about getting them to machine them for you. The standard way I have seen for a LCD is to cut the hole big enough for the screen but smaller then the edge. that way the LCD mounts tight to the inside and there is no gap. The next step is to bevel the edge all the way around so you don't have a 1/8" step . In a plastic enclosure this is quite easy to do by hand if its a 1 off. All you need is a razor knife and a flat file.
First cut the hole close with the razor knife(or saw). You could drill a series of holes first or use a dremel type tool to cut this.
Second, use the flat file to clean the hole and strighten the edges. Keep checking the fit and be patient. Then you can file on an angle to get the bevel.





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