Jerry as I understand the diode type laser will not cut copper sheet. I believe you'll need a fiber laser (different and much more $). However, the good news is if you had the more robust router type machine I believe you could cut .040 to .060" copper sheet using a small carbide tipped bit. The key (as I'm sure I don't need to tell you but will post for the benefit of others) is to lower the Z axis very little with each pass...maybe less than .1mm and possibly only around .01mm if you can manage it and to secure the copper to the worktable or spoilboard very well. Because of backlash in the lead screw and brass nut, most of these will not predictably handle .01mm but .1 to something like .05mm is usually okay. I believe I would use a 1/16" bit (two cutting flutes) and I'd limit the motor speed to 10,000 rpm or maybe less. I don't own one...I rely on a friend to help...but there are programs which will let you set up the shape, depth, etc you need and just walk away and do something else. You can see the image below. That is a fiberglass mold I made and then routed out the radio piece cutout. Using the program it takes about 20~25 minutes to make that cutout, and that's okay. You don't want to take bites that are too large. Once I was happy with the cutout, I made a silicone mold and from that I can make duplicates.
~ Jon 1952 1/2 ton with 1959 235 | T5 with 3.07 rear end